從完全圖中的獨立邊集構造法建構有容錯能力的群試設計

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國立高雄大學應用數學系

碩士論文

Error-correcting pooling designs constructed from

matchings of a complete graph

從完全圖中的獨立邊集構造法建構有容錯能力的群試設計

研究生：呂政和撰

指導教授：張惠蘭博士

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Error-correcting pooling designs

constructed from matchings of a

complete graph

by

Cheng-Huo Lu

Advisor

Huilan Chang

Department of Applied Mathematics

National University of Kaohsiung

Kaohsiung, Taiwan 811, R.O.C.

November 2018

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誌謝

回想起剛來到高雄大學，原本人生地不熟，從我修第一門課開始就是張惠蘭教授和旗下的同學們相互提攜、伴我度過這快樂的 2 年又 3 個月。會選擇惠蘭教授當我的指導教授，是要感謝郭岳承教授，看出我的性向適合往組合數學發展。我大學時，系上並沒有組合數學的課程，惠蘭教授的上課方式非常適合我，會一步一步慢慢教，我遇到不懂的時候就發問，惠蘭教授會逐一解答，不會一味的趕進度，因此我學習得很快。而同師門的同學們幾乎每天和我一起討論功課、一起吃飯，讓我學習之路不孤單。系上的老師人都很好，個性溫和不擺架子，不論是在修課或是我當助教時對我的指導，都為我的數學實力打下基礎。因此，最感謝的就是惠蘭教授、曾經教導過我的師長們和所有曾經陪伴過我的同學們。在口試前的兩個星期之間，感謝碩士班和大學部的學弟們幫我排練，並提出優化投影片排版方面的各種建議。不擅於作簡報的我雖然在學校排練了 4 次還是覺得不夠，因為內容很多而且又是連貫的一整套理論，很難刪減，為了在限時 30 分鐘之內講完 50 多頁的投影片，又為了表達出來的詞句必須精準到位、能夠通過數學教授的嚴格檢驗，報告的時候必須全神貫注、字字斟酌，所以排練時還沒有做到報告該有的流暢度。而且學弟們也點出我因為太緊張而緊盯著投影幕，沒有看到台下觀眾們的反應。這時我想出三種解決辦法多管齊下：打出逐字稿、錄音複核並且每天晚上在家裡排練。事後證明這三種方法非常有效，並且也感謝我的爸爸媽媽，雖然聽不懂我在報告什麼內容但還是每天耐心的當我的臨時聽眾。非常感謝王彩蓮教授和鄭斯恩教授擔任我的口試委員。斯恩教授非常仔細地審閱我的論文，幫我揪出很多我沒注意到的小細節，讓論文更加精緻完備；感恩彩蓮教授，在我作論文報告的時候專注地聆聽，並且當我報告到每個段落的重點時微笑的看著我並且對著我輕輕點頭回應，讓我更加有自信，將整場報告從容不迫、不疾不徐地講完。在提問的時候，因為問題太複雜，我在某一題的提問中卡住，更感恩三位口試委員都循序漸進地引導我先從簡單的問題開始發想，再延伸到原本複雜的問題，順利把該問題解決。我在口試前總共排練了十次以上，到最後一次時都還沒有達到完全理想，所幸口試當天臨場把狀況發揮得淋漓盡致，作出一個(至少是我自認)完美的報告，順利通過口試。原本懷著忐忑不安的心情參加口試，現在才能深刻體會，下了多少功夫，就有多少勝算。即將畢業了，從家人、師長、同學(學長、學弟、學妹)到口試委員，感恩天時地利人和，眾緣和合才能讓我順利拿到碩士學位。在高雄大學，一切都是美好的，這份情，這段憶，必將永生難忘。呂政和謹誌於國立高雄大學應用數學系中華民國一百零七年十一月

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從完全圖中的獨立邊集構造法建構有容錯能力的群試設計

指導教授張惠蘭教授國立高雄大學應用數學系學生呂政和國立高雄大學應用數學系

摘要

傳統群試的模型是：在 n 個元素中，最多只有 d 個壞的元素，我們想要用群試的方法以最 少的測驗次數來找出所有壞的元素。在不可調節的演算法中，又叫做池設計，我們必須一次把所有的測驗全部準備好，然後同步一起測。池設計在群試設計的領域中扮演重要地位。有很多關於血液測試、品質管控和基因檢測等方面的應用。 我們可以用(d;z)-分離矩陣來進行有容錯能力的池設計。一個二元矩陣，如果對任意 1 行 和 d 行都存在 z 列使得 1 行底下放 1、d 行底下都放 0，就叫做(d;z)-分離矩陣。一個(d;z)-分離 矩陣，如果當 z1 大於 z 時它不再是(d;z1)-分離矩陣，就叫做完全(d;z)-分離矩陣。令 M(m,m,d) 是一個二元矩陣，列代表所有的 K2m裡的 d-獨立邊集、行代表所有的 K2m裡的 m-獨立邊集(也 就是完美獨立邊集)，這個矩陣在第 i 列第 j 行的地方放 1 若且唯若第 i 個 d-獨立邊集被第 j 個 m-獨立邊集包住。 在本論文中，我們研究池設計 M(m,m,d)並且求它的容錯能力。我們的研究結果分成三個 部分： (1)當 m-d=1 時，M(m,m,d)是完全(d;z)-分離矩陣其中 z=m；當 m-d=2 時，M(m,m,d) 是完全(d;z)-分離矩陣其中 z=(𝑚₂)-d。

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(2)當 m≥d+3 且 1≤d≤ ⌊𝑚₂⌋時，M(m,m,d)是完全(d;z)-分離矩陣其中 z=2𝑑_。 (3)當 m≥d+3 且⌊𝑚₂⌋<d<m 時，M(m,m,d)是完全(d;z)-分離矩陣其中

z=O(42𝑑−𝑚3 ⋅ 3𝑚−𝑑_)。

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Error-correcting pooling designs

constructed from matchings of a complete

graph

Advisor: Huilan Chang

Department of Applied Mathematics National University of Kaohsiung

Student: Cheng-Huo Lu Department of Applied Mathematics

National University of Kaohsiung

ABSTRACT

The model of classical group testing is: given n items, at most d of them are defective, we want to find out all of the defectives using the minimum number of tests that are applied on a subset of these items. In a nonadaptive algorithm, also called a pooling design, one must decide all tests before any testing occurs. Pooling designs play an important role in the field of group testing. There are many applications such as blood testing, quality controlling and DNA screening.

Error-correcting pooling design uses (d; z)-disjunct matrices. A binary matrix M is called (d; z)-disjunct if given any d + 1 columns of M with one designated, there are z rows intersecting the designated column and none of the other d columns. A (d; z)-disjunct matrix is called completely (d; z)-disjunct if it is not (d; z1)-disjunct whenever z1 > z. Let M (m, m, d) be the 01-matrix whose rows are indexed by all

d-matchings on K2m and whose columns are indexed by all perfect matchings on

K2m. M (m, m, d) has a 1 in row i and column j if and only if the i-th d-matching is contained in the j-th m-matching.

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In this thesis, we study a pooling design M (m, m, d) and find its corresponding error-tolerance capabilities. Our results are divided into three parts: when m_{− d = 1} (resp. m _{− d = 2), M(m, m, d) is completely (d; z)-disjunct with z = m (resp.}

z = (m₂) − d); when m ≥ d + 3 and 1 ≤ d ≤ ⌊m/2⌋, M(m, m, d) is completely

(d; 2d_{)-disjunct; when m} _{≥ d + 3 and ⌊m/2⌋ < d < m, M(m, m, d) is completely}

(d; z)-disjunct where z = O(4(2d−m)/3_{· 3}m−d_).

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1 Introduction

Group testing was originally introduced by Robert Dorfman [3] as a potential ap-proach to economically screen soldiers with syphilis during World War II and the combinatorial group testing was first studied by Li [11]. There are many applications and variations of group testing such as blood testing, chemical leak testing, codes and DNA screening (see Du and Hwang [4, 5]).

The idea of group testing is a procedure that breaks up the task of identifying certain objects into tests on groups of items, rather than on individual ones. The model of classical group testing is: given n items, at most d of them are defective, we want to find out all of the defectives using tests that are applied on a subset of these items. In this thesis we focus on classical group testing.

A group test, also called a pool, is a subset of items. The outcome of a test is “positive” (denoted by 1) if at least one defective is tested. The outcome of a test is “negative” (denoted by 0) if no defective is tested.

Two types of group testing algorithms are often investigated: A sequential

algo-rithm conducts tests one by one and allows a later test to use the outcomes of all

previous tests. In a nonadaptive algorithm, also called a pooling design, one must decide all tests before any testing occurs.

A renewed interest of group testing has developed due to its applications in com-putational molecular biology (Balding et al., 1996 [2]; Farach et al., 1997 [8]). Since biological experiments are more time-consuming, nonadaptive algorithms are more applicable.

A nonadaptive algorithm is usually represented by a binary matrix M with columns indexed by items and rows indexed by tests, where a 1-enrty in Mij represents that

the i-th test contains the j-th item, and a 0-enrty in Mij represents that the i-th test

does not contain the j-th item.

1.1 Error-correcting pooling design

A binary matrix M is called d-disjunct if for any d + 1 columns of M with one designated, there is a row intersecting the designated column and none of the other

d columns. Disjunct matrices have a pivotal role in designing nonadaptive group

testing algorithms [5, Chapter 2]. To see this, given any good item a and a d-set P containing all defective items with a /_{∈ P , let C}0 be indexed by a and C1, C2, ..., Cdbe

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intersecting C0 but none of C1, C2, ..., Cd; thus, there is a test containing a but none

of items in P . So, a appears in a test answering 0. On the other hand, given any defective item b, any test containing b must answer 1. Therefore, an item is a good item if and only if there is a test answering 0 and containing it, thus after deleting all items appearing in any test answering 0, the remaining items are all defectives.

Biological experiments are known for producing erroneous outcomes. Therefore, it would be beneficial for pooling designs to have tolerance capability. An error-correcting pooling design is a nonadaptive group testing with at most e answers of tests “telling a lie” (test failed). Error-correcting pooling design uses (d; z)-disjunct matrices. A binary matrix M is called (d; z)-disjunct if given any d + 1 columns of M with one designated, there are z rows intersecting the designated column and none of the other d columns. A (d; 1)-disjunct matrix is simply d-disjunct. A matrix

M is (d; z)-disjunct implies that M is (d′; z′)-disjunct for all d′ ≤ d and z′ ≤ z. A (d; z)-disjunct matrix is called completely (d; z)-disjunct if it is not (d; z1)-disjunct whenever z1 > z.

A (d; z)-disjunct matrix is_{⌊(z − 1)/2⌋-correcting [6]. That is to say, an} error-correcting pooling design algorithm is valid whenever there are at most _{⌊(z − 1)/2⌋} answers of tests failed. To see this, given any good item a and a d-set P containing all defective items with a /_{∈ P , let C}0 be indexed by a and C1, C2, ..., Cd be indexed

by items in P . According to the (d, z)-disjunct property of M , there are z rows intersecting C0 but none of C1, C2, ..., Cd; thus, there are z tests containing a but

none of items in P . So, a appears in z tests answering 0. After considering at most

⌊(z − 1)/2⌋ failed tests, a appears in at least ⌊(z − 1)/2⌋ + 1 tests answering 0. On

the other hand, given any defective item b, any test containing b must answer 1. After considering at most _{⌊(z − 1)/2⌋ failed tests, b appears in at most ⌊(z − 1)/2⌋} tests answering 0. Therefore, an item is a good item if and only if there are at least _{⌊(z − 1)/2⌋ + 1 tests answering 0 and containing it, thus after deleting all items} appearing in at least _{⌊(z − 1)/2⌋ + 1 test answering 0, the remaining items are all} defectives.

1.2 A brief survey of error-correcting pooling design

con-structed by inclusion of non-perfect matchings

A matching in a graph is a set of edges without common vertices. A matching of size l(i.e. it has l edges) is called an l-matching. A perfect matching is a matching

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which matches all vertices of the graph; that is, every vertex of the graph is incident to exactly one edge of the matching.

Many famous pooling designs have been constructed from mathematical structures by the containment matrix method (see [1], [6], [7] and [12]). Subsequently, some improvements on error-tolerance capability have been proposed by considering the “intersecting relations” (see [9], [10] and [14]).

D’yachkov et al.(2007) [7] discussed three types of inclusion matrices (subset, subspace and sequence) and exhibited their disjunct properties and error-tolerance capabilities. Another type of inclusion matrices is constructed by matchings as fol-lows:

Definition 1.1. For 1 _{≤ d < k ≤ m, let M(m,k,d) be the 01-matrix whose rows are}

indexed by all d-matchings on K2m, and whose columns are indexed by all k-matchings on K2m. M (m, k, d) has a 1 in row i and column j if and only if the i-th d-matching is contained in the j-th k-matching.

Theorem 1.2. ([15, Theorem 6]) For 1 _{≤ d < k ≤ m, M(m, k, d) is a g(m, d) ×}

g(m, k) d-disjunct matrix with row weight g(m− d, k − d) and column weight (k_d), where g(m, t) =(2m

2t )_(2t)!

2t_t!.

In graph theory, a distance-regular graph is a regular graph such that for any two vertices u and v, the number of vertices at distance j from u and at distance k from v depends only upon j, k and the distance between u and v. One of the distance-regular graphs is Johnson graph. Define [n] =_{{1, 2, 3, ..., n} and denote by} (S

t

)

the family of all t-subsets of S. The Johnson graph J(n, t) is defined on([n]

t

)

such that two vertices

u and v are adjacent if and only if |u ∩ v| = t − 1. Many error-correcting pooling

designs can associate with some distance-regular graphs. Bai et al. [1] studied the error-tolerance of matrices constructed from some structures of the Johnson graph.

Let G = (V, E) be a connected graph. An l-subset C of V is said to be a t-clique

of G with size l if any two distinct vertices in C are at distance t. A t-clique with size l of Johnson graph J (n, t) is simply a collection of l disjoint t-subsets of [n]. Bai et

al. [1] studied matrices constructed from t-cliques with varies sizes in Johnson graph

J (n, t). Let M (A, B) denote the (A, B)-th entry of a matrix.

Definition 1.3. Given positive integers d < k and kt _{≤ n, let J(n, t, d, k) be the}

binary matrix with rows (resp. columns) indexed by t-cliques with size d (resp. k) of J (n, t) such that M (A, B) = 1 if and only if A⊆ B.

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Bai et al. [1] analyzed the error-tolerance capability of J(n, t, d, k) and compared with other pooling designs that are associated with distance-regular graphs.

Theorem 1.4. ([1, Corollary 3.5]) Let 1 _{≤ s ≤ d < k and (k + 1)t ≤ n. Then}

J (n, t, d, k) is completely (s; z)-disjunct with z =(k_d−s_−s).

The case when t = 2 of the Theorem is the following Corollary.

Corollary 1.5. Let 1 _{≤ s ≤ d < k and k < m. Then M(m, k, d) is completely}

(s; z)-disjunct with z =(k_d−s_−s).

So, we can see that the study of error-correcting pooling design constructed by inclusion of matchings has been well studied if k < m, namely, columns are indexed by matchings of K2m which are not perfect.

In this thesis, we consider matrices M (m, m, d) and analyze their error-tolerance capabilities.

1.3 Preview of the thesis

Now the error-tolerance of M (m, m, d) has not been well studied yet. In this thesis we want to study the optimal error-tolerance capability of M (m, m, d).

In Section 2, we study the error-tolerance capabilities of pooling designs con-structed by M (m, m, d) from combinatorial structures proposed by Ngo and Du [15]. We introduce some ideas to get started. In Section 3 we present some Lemmas that are used to narrow down the possibilities of an optimal pattern. We then give main results about the error-tolerance of the pooling design M (m, m, d) when m_{− d ≤ 2}

or d _{≤ ⌊m/2⌋. In Section 4 we give a conjecture about the error-tolerance of the}

pooling design M (m, m, d) when m_{− d ≥ 3 and ⌊m/2⌋ < d < m and give a table of} our results in summary.

2 Preliminaries and Main Ideas

In this section, we introduce the main ideas used to deal with our study, that is, the error-tolerance of M (m, m, d).

Definition 2.1. For any d + 1 columns C0, C1, ..., Cd of a matrix, we call the rows

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A vertex cover of a graph is a set of vertices such that each edge of the graph is incident to at least one vertex of the set. A vertex cover of size d is called a d-cover. Now we describe how this question have been studied. To study the error-tolerance of M (m, m, d), we use the following technique.

First, we give a property which is straight forward by definition.

Property 2.2. In the question of finding the number z such that M is completely (d; z)-disjunct, we only need to seek a set of d+1 columns with one designated such that

the combination of these d + 1 columns produces the least number of error-correcting rows.

For any d + 1 perfect matchings C0, C1, ..., Cd of K2m, every d-matching D of K2m, where D _{⊆ C}0 but D * Ci for i = 1, 2, ..., d, corresponds to a row that intersects

column C0 but none of columns C1,...,Cd.

Observe that for each i_{∈ [d], C}0∪ Ci is a loop-less multi-graph which is 2-regular. C0∪ Ci consists of cycles with even lengths. Moreover, C0 ̸= Ci implies that C0∪ Ci must have a cycle of length at least 4; consequently, _|C0 \ Ci| ≥ 2, ∀i ∈ [d]. For each i _{∈ [d], choose arbitrary Ei} _{⊆ C}0 \ Ci so that |Ei| = 2. Let G[C0; E1, ..., Ed]

be the graph with V (G[C0; E1, ..., Ed]) = C0 and E(G[C0; E1, ..., Ed]) ={E1, ..., Ed} (multiset). Then, G[C0; E1, ..., Ed] is a multi-graph having m vertices and d edges.

Any d-subset D of C0 such that D ∩ Ei ̸= ∅, for i = 1, 2, ..., d contributes a row intersecting column C0 but none of C1, ..., Cd. Note that D is nothing but a d-cover of G[C0; E1, ..., Ed], so every d-cover D of G[C0; E1, ..., Ed] contributes a row intersecting

column C0 but none of C1, ..., Cd. See Figure 1 for an example.

Then the following property can be straight forward obtained by the above de-scription.

Property 2.3. For any d + 1 columns C0, C1, ..., Cd, let Ei be a 2-subset of C0\ Ci

for 1_{≤ i ≤ d. Every d-cover in G[C}0; E1, ..., Ed] corresponds to a d-matching D such that D _{⊆ C}0 and D* Ci, ∀i ∈ [d].

The following Lemma and Theorem are done by Ngo and Du[15].

Lemma 2.4. ([15, Lemma 10]) Given integers m > d ≥ 1 and any labelled simple

graph G with |V (G)| = m and |E(G)| = d, the number of d-covers of G is at least

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By Ngo and Du’s argument, it is checked that M (m, m, d) is (d, d + 1)-disjunct by showing that for any d + 1 columns C0, C1, ..., Cdof a matrix, the number of

error-correcting row is at least d+1. By Property 2.3, for any d+1 columns C0, C1, ..., Cd, let Ei be a 2-subset of C0\Ci for 1≤ i ≤ d, every d-cover in G[C0; E1, ..., Ed] corresponds

to an error-correcting row for (C0,{C1, ..., Cd}). If G[C0; E1, ..., Ed] has an multi-edge,

then let G be any simple graph with m vertices and d edges that contains the graph obtained from removing all multi-edges of G[C0; E1, ..., Ed]. Then note that a d-cover

of G is a d-cover of G[C0; E1, ..., Ed], so the number of d-covers of G[C0; E1, ..., Ed]

is equal to or more than the number of d-covers of G. By Lemma 2.4, every simple graph G with _{|V (G)| = m and |E(G)| = d has at least d + 1 d-covers of G, so the} proof of the following Theorem is completed.

Theorem 2.5. ([15, Theorem 11]) For 1≤ d < m, M(m, m, d) is (d; d + 1)-disjunct.

Based on the above idea and argument, we continue to study forward. For any

d + 1 columns C0, C1, ..., Cd, when |C0\ Ci| = 2 for all i ∈ [d], let Ei = C0\ Ci, if a d-subset R is not a d-cover of G[C0; E1, ..., Ed], then there is at least one edge whose

two vertices are not in the d-subset R. Assume that this edge is just the 2-set C0\ Ci for some i _{∈ [d]. Then R ⊆ Ci} and thus R does not contribute an error-correcting row for (C0,{C1, ..., Cd}). We have the following property.

Property 2.6. For any d + 1 columns C0, C1, ..., Cd, if |C0 \ Ci| = 2 for all i ∈ [d],

let Ei = C0 \ Ci. Then every d-matching D such that D ⊆ C0 and D * Ci, ∀i ∈ [d]

corresponds to a d-cover in G[C0; E1, ..., Ed].

The above two Property 2.3 and 2.6 can be combined to form this one.

Lemma 2.7. For any d + 1 distinct perfect matchings C0, C1, ..., Cd of K2m, if |C0\

Ci| = 2 for all i ∈ [d], let Ei = C0\Ci. Then the number of d-covers of G[C0; E1, ..., Ed] is exactly the number of error-correcting rows produced by (C0,{C1, ..., Cd}).

The following two Lemmas are about the case when G[C0; E1, ..., Ed] is not a

simple graph and the case when |C0\ Ci| > 2 for some i ∈ [d].

Lemma 2.8. For any d + 1 columns C0, C1, ..., Cdwith |C0\Ci| = 2 for all i ∈ [d], let

Ei = C0\ Ci. If G[C0; E1, ..., Ed] is not a simple graph, then there exist another d + 1 columns C0, C1′, ..., Cd′ satisfying |C0 \ Ci′| = 2 for all i ∈ [d] such that in M(m, m, d) the number of error-correcting rows produced by (C0,{C1, ..., Cd}) is equal to or larger than the number of error-correcting rows produced by (C ,{C′, ..., Cd}).′

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Proof. For any d+1 columns C0, C1, ..., Cd, if G[C0; E1, ..., Ed] is not a simple graph,

let us consider another set of d+1 columns_{C0, C1′, ..., Cd} such that {E′ 1, E2, ..., Ed} ⊆ {E′

1, E2′, ..., Ed} where E′ i′ = C0\ Ci′ and all Ei′ are all distinct. Then since a d-cover

of G[C0; E1′, ..., Ed′] is a d-cover of G[C0; E1, ..., Ed], by Lemma 2.7 and Property 2.3

we can see that the number of error-correcting rows produced by (C0,{C1′, ..., Cd}) is′ equal to or less than the number of error-correcting rows produced by (C0,{C1, ..., Cd}).

Lemma 2.9. For any d + 1 columns C0, C1, ..., Cd, if |C0\ Ci| > 2 for some i ∈ [d],

then there exist another d + 1 columns C0, C1′, ..., Cd′ satisfying |C0 \ Ci| = 2 for′

all i _{∈ [d] such that in M(m, m, d) the number of error-correcting rows produced}

by (C0,{C1, ..., Cd}) is equal to or larger than the number of error-correcting rows produced by (C0,{C1′, ..., Cd}).′

Proof. Let Ei be a 2-subset of C0 \ Ci for 1 ≤ i ≤ d. By Property 2.3 we can see

that the number of d-covers of G[C0; E1, ..., Ed] is equal to or less than the number of

error-correcting rows produced by (C0,{C1, ..., Cd}). We consider the following two cases.

Case 1, if G[C0; E1, ..., Ed] is a simple graph, let us consider another set of d + 1

columns _{C0, C1′, ..., Cd} such that C′ 0 \ Ci′ = Ei, then since |C0 \ Ci′| = 2 for all i ∈ [d], by Lemma 2.7 the number of d-covers of G[C0; E1, ..., Ed] is equal to the

number of error-correcting rows produced by (C0,{C1′, ..., Cd}). So the number of′ error-correcting rows produced by (C0,{C1′, ..., Cd}) is equal to or less than the number′ of error-correcting rows produced by (C0,{C1, ..., Cd}).

Case 2, if G[C0; E1, ..., Ed] is not a simple graph, let us consider another set of d+1

columns_{C0, C1′, ..., Cd} satisfying |C′ 0\Ci| = 2 for all i ∈ [d] such that {E′ 1, ..., Ed} ⊆ {E′

1, ..., Ed} where E′ i′ = C0\ Ci′ and all Ei′ are all distinct. Then since |C0\ Ci| = 2′ for all i _{∈ [d], by Lemma 2.7 the number of d-covers of G[C}0; E1′, ..., Ed′] is equal to

the number of error-correcting rows produced by (C0,{C1′, ..., Cd}). Since a d-cover of′ G[C0; E1′, ..., Ed′] is a d-cover of G[C0; E1, ..., Ed], we can see that the number of

error-correcting rows produced by (C0,{C1′, ..., Cd}) is equal to or less than the number of′ error-correcting rows produced by (C0,{C1, ..., Cd}).

By Property 2.2, Lemma 2.8 and Lemma 2.9, we can ignore the case that G[C0; E1, ..., Ed]

is not a simple graph and the case that_|C0\ Ci| > 2 for some i ∈ [d] in the question of finding number z such that M (m, m, d) is completely (d; z)-disjunct.

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By now we find a bridge across the problem of completely (d; z)-disjunct of

M (m, m, d) and the d-covers of G[C0; E1, ..., Ed].

3 The Error-tolerance of M (m, m, d)

For any graph G with m vertices and d edges, there exist d + 1 perfect matchings

C0, C1, ..., Cdsuch that G[C0; E1, ..., Ed] is isomorphic to G. Then according to Lemma 2.7, Lemma 2.9 and Lemma 2.8, our goal turns to identify one simple graph with m vertices and d edges that has the minimum number of d-covers. A pattern is a simple graph with m vertices and d edges. A pattern is optimal if it has the minimum number of d-covers among all simple graphs of m vertices and d edges.

Some patterns have distinct numbers of d-covers, see Figure 2.

3.1 Narrow down the possibilities of an optimal pattern

Consider all the patterns, we want to find which is optimal, and to construct how the pattern looks like. We introduce a strategy.

Definition 3.1. For a pattern G, we use c(G) to denote the number of d-covers of

G. For two disjoint subsets S, S′ of edges of G, use c(G, S, S′) to denote the number

of d-subset D of V (G) such that D intersects every edge in S but no edge in S′.

Lemma 3.2. Suppose that G and G′ are two patterns with exactly d _{− 1}

com-mon edges e1, e2, ..., ed−1, E(G) \ E(G′) = {ed} and E(G′) \ E(G) = {e′d}. If c(G,{e1, e2, ..., ed−1}, {ed}) < c(G′,{e1, e2, ..., ed−1}, {e′d}), then c(G′) < c(G). Be-sides, if c(G,_{e1, e2, ..., ed−1}, {ed}) ≤ c(G′,{e1, e2, ..., ed−1}, {e′d}), then c(G′)≤ c(G).

Proof. Notice that the number of d-subsets of V (G) equals to c(G, ϕ,_{e1}) +

c(G,{e1}, {e2})+...+c(G, {e1, e2, ..., ed−2}, {ed−1})+c(G, {e1, e2, ..., ed−1}, {ed})+c(G)

and similarly for G′. Since by using the identity mapping f : V (G) _{→ V (G}′), every

d-subset D of V (G) such that D intersects every edge in S but no edge in S′ corre-sponds to a d-subset f (D) of V (G′) such that f (D) intersects every edge in S but no edge in S′, and vice versa, we have c(G, ϕ,_{e1}) = c(G′, ϕ,{e1}), c(G, {e1}, {e2}) = c(G′,{e1}, {e2}),..., c(G, {e1, e2, ..., ed−2}, {ed−1}) = c(G′,{e1, e2, ..., ed−2}, {ed−1}), so

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We illustrate an example of how this strategy is operated in showing a pattern G is not optimal by considering another pattern G′, where m = 6 and d = 3. See Figure 3. Now, we introduce some lemmas used to prove the main result, it is a little bit lengthy but they are all indispensable. In Figure 4, we give some pictures to illustrate these Lemmas for easy reading; in Figure 5, we show relations between lemmas. Definition 3.3. A cycle is a sequence of distinctive adjacent vertices that begins and

ends at the same vertex. A path is a sequence of distinctive vertices connected by edges. A tree is a connected simple graph without cycles. Denote by Cn the cycle with n vertices. Denote by Pn the path with n vertices.

Lemma 3.4. For m _{≥ d + 3, if G = Hc}_{∪ P}1 ∪ H, where Hc is a component that is

not acyclic, then there exists another pattern G′ such that c(G′)≤ c(G).

Proof. Let C be a cycle in Hc and let v1, v2, ..., v|C|+1, ..., vm be an ordering of

vertices of V (G) such that v1 is an isolated vertex of G and C = v2v3...v|C|+1. Assume that E(G) =_{e1, e2, ..., ed} where ed={v2, v|C|+1}. Let G′ be a graph with V (G′) = V (G) and E(G′) ={e1, e2, ..., e′d} where e′d={v1, v|C|+1}.

We claim that c(G,_{e1, e2, ..., ed−1}, {ed}) ≤ c(G′,{e1, e2, ..., ed−1}, {e′d}). We

con-sider the bijection f : V (G)_{→ V (G}′) with f (v2) = v1, f (v1) = v2, f (x) = x whenever x ̸= v1 and x ̸= v2. See Figure 6 for easy reading. Then the mapping D 7→ f(D) from(V (G) d ) to(V (G′) d ) is injective.

Let D be a d-subset of V (G) that intersects every edge in _{e1, e2, ..., ed−1} but

does not intersect edge ed={v2, v|C|+1}. We shall show that f(D) ⊆ V (G′) intersects every edge e_{∈ {e}1, e2, ..., ed−1} but does not intersect e′d ={v1, v|C|+1} in G′.

Since v2, v|C|+1 ∈ D, v/ 1 = f (v2), v|C|+1 = f (v|C|+1) /∈ f(D) and thus f(D) does not intersect e′_d. Consider that in G′ if e is not incident with v2 or v|C|+1, let e = {vj, vk}, then we have vj ∈ f(D) or vk ∈ f(D), since if vj, vk ∈ f(D), then vj/ = f−1(vj) /∈ D and vk = f−1(vk) /∈ D, and so {vj, vk} ∈ E(G) is not intersected by D,

a contradiction. Consider that in G′ if e is incident with v2, let e ={v2, vj}, then we have vj ∈ f(D) due to the reason that vj = f−1(vj)∈ D. Consider that in G′ if e is

incident with v_|C|+1, let e = _{v|C|+1, vj}, then we have vj ∈ f(D) due to the reason

that vj = f−1(vj) ∈ D. So f(D) ⊆ V (G′) intersects every edge e ∈ {e1, e2, ..., ed−1}

but does not intersect e′_d = {v1, v|C|+1} in G′. Hence c(G,{e1, e2, ..., ed−1}, {ed}) ≤

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Figure 4: A brief introduction of each Lemma

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Figure 6: The instruction of Lemma 3.4

Lemma 3.5. For m_{≥ d + 3, if G = Hc}_{∪ P}1∪ P1∪ H, where Hc is a component that

is not acyclic, then there exists a pattern G′ such that c(G′) < c(G).

Proof. Let C be a cycle in Hc and let v1, v2, ..., v|C|+2, ..., vm be an ordering of

vertices of V (G) such that v1 and v2 are isolated vertices of G and C = v3v4...v|C|+2. Assume that E(G) = _{e1, e2, ..., ed} where ed = {v3, v4}. Let G′ be a graph with V (G′) = V (G) and E(G′) ={e1, e2, ..., e′d} where e′d={v1, v2}.

First, we claim that c(G,_{e1, e2, ..., ed−1}, {ed}) ≤ c(G′,{e1, e2, ..., ed−1}, {e′d}). We

consider the bijection f : V (G) _{→ V (G}′) with f (v1) = v3, f (v2) = v4, f (v3) = v1, f (v4) = v2, f (x) = x whenever x ̸= vi for i = 1, 2, 3, 4. See Figure 7 for easy reading. Then the mapping D _{7→ f(D) from}(V (G)

d ) to (V (G′) d ) is injective.

does not intersect edge ed = {v3, v4}. We shall show that f(D) ⊆ V (G′) intersects every edge e_{∈ {e}1, e2, ..., ed−1} but does not intersect ed′ ={v1, v2} in G′.

Since v3, v4 ∈ D, v/ 1 = f (v3), v2 = f (v4) /∈ f(D) and thus f(D) does not intersect e′_d. Consider that in G′ if e is not incident with v3 or v4, let e = {vj, vk}, then we have vj ∈ f(D) or vk ∈ f(D), since if vj, vk ∈ f(D), then vj/ = f−1(vj) /∈ D and vk = f−1(vk) /∈ D, and so {vj, vk} ∈ E(G) is not intersected by D, a contradiction.

Consider that in G′ if e is incident with v3, let e ={v3, vj}, then we have vj ∈ f(D) due to the reason that vj = f−1(vj)∈ D. Consider that in G′ if e is incident with v4, let e = _{v4, vj}, then we have vj ∈ f(D) due to the reason that vj = f−1(vj) ∈ D.

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So f (D) _{⊆ V (G}′) intersects every edge e ∈ {e1, e2, ..., ed−1} but does not intersect e′_d={v1, v2} in G′. Hence c(G,{e1, e2, ..., ed−1}, {ed}) ≤ c(G′,{e1, e2, ..., ed−1}, {e′d}).

Second, we claim further that c(G,_{e1, e2, ..., ed−1}, {ed}) < c(G′,{e1, e2, ..., ed−1}, {e′d}).

We shall show that there exists a d-subset D′ _{⊆ V (G}′) intersecting every edge in

{e1, e2, ..., ed−1} but no edge in {e′d}, but f−1(D′) ⊆ V (G) does not intersect

ev-ery edge in _{e1, e2, ..., ed−1} but no edge in {ed}. Arbitrarily choosing one vertex

from each ei, i ̸= d, yields a cover of G′ − v1 − v2 of size at most d− 1. Since G′ − v1 − v2 has at least d + 1 vertices and exactly d − 1 edges, there exists a d-subset D′ _{⊆ V (G}′) that intersects every edge in {e1, ..., ed−1} but no edge in {e′d}

and v5 ∈ D/ ′. Since v2, v5 ∈ D/ ′, the edge {e4, e5} is not covered by f−1(D′) in G. So c(G,_{e1, e2, ..., ed−1}, {ed}) < c(G′,{e1, e2, ..., ed−1}, {e′d}). By Lemma 3.2,

c(G′) < c(G).

Lemma 3.6. For m ≥ d + 3, if G = T ∪ H, where T is a component that is a tree

but not a path, then there exists a pattern G′ such that c(G′) < c(G).

Proof. Let P be a maximal path in T and let v1, v2, ..., vi, ..., v|P |, ..., vm be an

or-dering of vertices of V (G) such that _{vi, v|P |+1} is the first branch of P from v1 to v|P | and P = v1v2...v|P |. Assume that E(G) ={e1, e2, ..., ed} where ed ={vi, v|P |+1}. Let G′ be a graph with V (G′) = V (G) and E(G′) ={e1, e2, ..., e′d} where e′d ={v1, v|P |+1}. First, we claim that c(G,_{e , e , ..., e −1}, {ed}) ≤ c(G′,{e , e , ..., e −1}, {e′d}). We

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consider the bijection f : V (G) _{→ V (G}′) with f (vk) = vi+1−k whenever 1 ≤ k ≤ i, f (vk) = vkwhenever i+1≤ k ≤ m. See Figure 8 for easy reading. Then the mapping D7→ f(D) from (V (G)_d ) to(V (G′)

d

)

is injective.

does not intersect edge ed={vi, v|P |+1}. We shall show that f(D) ⊆ V (G′) intersects

every edge e _{∈ {e}1, e2, ..., ed−1} but does not intersect e′d = {v1, v|P |+1} in G′. Since vi, v|P |+1 ∈ D, v/ 1 = f (vi), v|P |+1= f (v|P |+1) /∈ f(D) and thus f(D) does not intersect e′_d. We consider the following two cases.

Case 1: e =_{vk, vk+1} ∈ E(G′) with 1≤ k ≤ i − 1. Consider that in G′ if e is not incident with v1, then we have vk ∈ f(D) or vk+1 ∈ f(D), since if vk, vk+1 ∈ f(D),/

then vi+1−k = f−1(vk) /∈ D and vi−k = f−1(vk+1) /∈ D, and so {vi−k, vi−k+1} ∈ E(G)

is not intersected by D, a contradiction. Consider that in G′ if e =_{v1, v2}, then we have v2 ∈ f(D) due to the reason that vi−1 = f−1(v2)∈ D.

Case 2: e = _{vk, vl} ∈ E(G′) with i ≤ k < l ≤ m. Consider that in G′ if e is not incident with vi or v|P |+1, then we have vk ∈ f(D) or vl ∈ f(D), since if vk, vl ∈ f(D), then vk/ = f−1(vk) /∈ D and vl = f−1(vl) /∈ D, and so {vk, vl} ∈ E(G)

is not intersected by D, a contradiction. Consider that in G′ if e is incident with vi,

let e = _{vi, vj}, then we have vj ∈ f(D) due to the reason that vj = f−1(vj) ∈ D.

Consider that in G′ if e is incident with v_{|P |+1}, let e = _{{v|P |+1}, vj}, then we have vj ∈ f(D) due to the reason that vj = f−1(vj)∈ D.

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So f (D)_{⊆ V (G}′) intersects every edge e∈ {e1, e2, ..., ed−1} but does not intersect e′_d={v1, v|P |+1} in G′. Hence c(G,{e1, e2, ..., ed−1}, {ed}) ≤ c(G′,{e1, e2, ..., ed−1}, {e′d}).

We shall show that there exists a d-subset D′ _{⊆ V (G}′) intersecting every edge in

{e1, e2, ..., ed−1} but no edge in {e′d}, but f−1(D′) ⊆ V (G) does not intersect every

edge in_{e1, e2, ..., ed−1} but no edge in {ed}. For i ̸= d, arbitrarily choosing one vertex

from each ei that are not incident with v1or v|P |+1, and choosing the other vertex from each ei that are incident with v1 or v|P |+1, yields a cover of G′−e′dwithout choosing v1 and v_{|P |+1}of size at most d_{−1. Since G}′_−e′_dhas at least d+3 vertices and exactly d₋₁ edges, there exists a d-subset D′ _{⊆ V (G}′) that intersects every edge in {e1, ..., ed−1}

but no edge in_{e′_{d} and vi+1}∈ D/ ′. Since v1, vi+1∈ D/ ′, the edge{vi, vi+1} is not

cov-ered by f−1(D′) in G. So c(G,{e1, e2, ..., ed−1}, {ed}) < c(G′,{e1, e2, ..., ed−1}, {e′d}).

By Lemma 3.2, c(G′) < c(G).

Let H1 and H2 be two graphs with no common vertex and no common edge, the symbol H1∪ H2 denotes a graph G with V (G) = V (H1)∪ V (H2) and E(G) = E(H1)∪E(H2). The following Lemmas are about the deletion of some special pattern

G and leave some possibly optimal pattern G′ remains.

Lemma 3.7. For s_{≥ 3 and m ≥ d + 3, if G = Ps}_{∪ P}1∪ H and G′ = P2∪ Ps−1∪ H,

then c(G′) < c(G).

Proof. G = Ps ∪ P1 ∪ H. Let v1, v2, ..., vs+1, ..., vm be an ordering of vertices of

V (G) such that v1 is an isolated vertex of G and Ps = v2v3...vs+1. Assume that E(G) = {e1, e2, ..., ed} where ed = {v2, v3}. G′ = P2∪ Ps−1 ∪ H. Let e′d = {v1, v2}. We assume that V (G′) = V (G) and E(G′) ={e′d} ∪ (E(G) \ {ed}). Note that in G′,

P2 = v1v2 and Ps−1 = v3v4...vs+1.

consider the bijection f : V (G) _{→ V (G}′) with f (v2) = v1, f (v3) = v2, f (v1) = v3, f (x) = x whenever x̸= vi for i = 1, 2, 3. See Figure 9 for easy reading. Then the mapping D_{7→ f(D) from} (V (G) d ) to (V (G′) d ) is injective.

does not intersect edge ed = {v2, v3}. We shall show that f(D) ⊆ V (G′) intersects every edge e_{∈ {e}1, e2, ..., ed−1} but does not intersect ed′ ={v1, v2} in G′.

Since v2, v3 ∈ D, v/ 1 = f (v2), v2 = f (v3) /∈ f(D) and thus f(D) does not intersect e′_d. Consider that in G′ if e is not incident with v3, let e ={vj, vk}, then we have vj ∈ f (D) or vk ∈ f(D), since if vj, vk∈ f(D), then vj/ = f−1(vj) /∈ D and vk = f−1(vk) /∈

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D, and so {vj, vk} ∈ E(G) is not intersected by D, a contradiction. Consider that in G′ if e = _{v3, v4}, then we have v4 ∈ f(D) due to the reason that v4 = f−1(v4)∈ D. So f (D) _{⊆ V (G}′) intersects every edge e ∈ {e1, e2, ..., ed−1} but does not intersect e′_d={v1, v2} in G′. Hence c(G,{e1, e2, ..., ed−1}, {ed}) ≤ c(G′,{e1, e2, ..., ed−1}, {e′d}).

Let D′ = K ∪ {v3, v5, v6, ..., vs+1} where K is a cover of H of size d − s + 2. Then it is clear that D′ intersects every edge in _{e1, e2, ..., ed−1} but does not intersect e′_d = {v1, v2} in G′. H has exactly m− s − 1 vertices. Since m ≥ d + 3, H has at least d_{− s + 2 vertices. Since H has exactly d − s + 1 edges, arbitrarily choosing one} vertex from each e_{∈ E(H) yields a cover of H of size at most d − s + 1. Thus H has} a cover of size d_{− s + 2. Since v}2, v4 ∈ D/ ′, the edge{v3, v4} is not covered by f−1(D′) in G. So c(G,_{e1, e2, ..., ed−1}, {ed}) < c(G′,{e1, e2, ..., ed−1}, {e′d}). By Lemma 3.2,

c(G′) < c(G).

Lemma 3.8. For s, t≥ 3 and m ≥ d+3, if G = Ps∪Pt∪H and G′ = P2∪Ps+t−2∪H,

then c(G′)≤ c(G).

Proof. G = Ps∪ Pt∪ H. Let v1, v2, ..., vs, vs+1, ..., vs+t, vs+t+1, ..., vm be an ordering

of vertices of V (G) such that Ps = v1v2...vs and Pt = vs+1vs+2...vs+t. Assume that E(G) ={e1, e2, ..., ed} where ed ={v2, v3}. G′ = P2∪ Ps−1∪ H. Let e′d ={vs, vs+1}.

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P2 = v1v2 and Ps+t−2 = v3v4...vs+t.

We claim that c(G,_{e1, e2, ..., ed−1}, {ed}) ≤ c(G′,{e1, e2, ..., ed−1}, {e′d}). We

con-sider the bijection f : V (G) _{→ V (G}′) with f (vi) = vs+3−i whenever 1 ≤ i ≤ s + 2, f (vi) = vi whenever s+3≤ i ≤ m. See Figure 10 for easy reading. Then the mapping D7→ f(D) from (V (G)_d ) to(V (G′)

d

)

is injective.

does not intersect edge ed = {v2, v3}. We shall show that f(D) ⊆ V (G′) intersects every edge e _{∈ {e}1, e2, ..., ed−1} but does not intersect e′d = {vs, vs+1} in G′. Since v2, v3 ∈ D, vs+1/ = f (v2), vs = f (v3) /∈ f(D) and thus f(D) does not intersect e′d. We

consider the following three cases.

Case 1: e = _{vj, vk} ∈ E(G′) with j < k ≤ s + 2. Consider that in G′ if

e is not incident with vs and vs+1, then we have vj ∈ f(D) or vk ∈ f(D), since

if vj, vk ∈ f(D), then vs+3/ −j = f−1(vj) /∈ D and vs+3−k = f−1(vk) /∈ D, and so {vs+3−k, vs+3−j} ∈ E(G) is not intersected by D, a contradiction. Consider that in G′

if e =_{vs₋₁, vs}, then we have vs₋₁ ∈ f(D) due to the reason that v4 = f−1(vs−1)∈ D.

Consider that in G′ if e =_{vs+1, vs+2}, then we have vs+2 ∈ f(D) due to the reason

that v1 = f−1(vs+2)∈ D.

Case 2: e =_{vs+2, vs+3} ∈ E(G′). Since v1 ∈ D, f(v1) = vs+2 ∈ f(D) and thus e

is intersected by f (D).

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vk ∈ f(D), since if vj, vk ∈ f(D), then vj/ = f−1(vj) /∈ D and vk = f−1(vk) /∈ D, and

so _{vj, vk} ∈ E(G) is not intersected by D, a contradiction.

So f (D)_{⊆ V (G}′) intersects every edge e∈ {e1, e2, ..., ed−1} but does not intersect e′_d={vs, vs+1} in G′. Hence c(G,_{e1, e2, ..., ed−1}, {ed}) ≤ c(G′,{e1, e2, ..., ed−1}, {e′d}).

By Lemma 3.2, c(G′)≤ c(G).

Lemma 3.9. For any m, d, if G = P4∪ H and G′ = C3∪ P1∪ H, then c(G′) = c(G).

Proof. G = P4 ∪ H. Let v1, v2, v3, v4, ..., vm be an ordering of vertices of V (G)

such that P4 = v1v2v3v4. Assume that E(G) = {e1, e2, ..., ed} where ed = {v3, v4}. G′ = C3 ∪ P1 ∪ H. Let e′d ={v1, v3}. We assume that V (G′) = V (G) and E(G′) = {e′_{d} ∪ (E(G) \ {ed}). Note that in G}′_{, C}

3 = v1v2v3 and P1 = v4.

We claim that c(G,_{e1, e2, ..., ed−1}, {ed}) = c(G′,{e1, e2, ..., ed−1}, {e′d}). We

con-sider the bijection f : V (G)_{→ V (G}′) with f (v1) = v4, f (v3) = v1, f (v4) = v3, f (x) = x whenever x̸= vi for i = 1, 3, 4. See Figure 11 for easy reading. Then the mapping

D7→ f(D) from (V (G)_d ) to(V (G′) d

)

is injective.

Let D be a d-subset of V (G). We shall show that D _{⊆ V (G) intersects} ev-ery edge in _{e1, e2, ..., ed−1} but does not intersect edge ed = {v3, v4} if and only if f (D) ⊆ V (G′) intersects every edge in {e1, e2, ..., ed−1} but does not intersect edge e′_d={v1, v3}.

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Since v1 = f (v3) and v3 = f (v4), D does not intersect {v3, v4} if and only if f(D) does not intersect _{v1, v3}.

D ⊆ V (G) covers H in G if and only if f(D) = D′ ⊆ V (G′) covers H in G′.

Further, D _{⊆ V (G) intersects every edge in {e}1, e2, ..., ed−1} if and only if D

cov-ers H and v2 ∈ D; D′ ⊆ V (G′) intersects every edge in {e1, e2, ..., ed−1} if and

only if D′ covers H and f (v2) = v2 ∈ D′. Therefore, according to the bijection f , D ⊆ V (G) intersects every edge in {e1, e2, ..., ed−1} if and only if f(D) ⊆ V (G′)

intersects every edge in _{e1, e2, ..., ed−1}. This implies c(G, {e1, e2, ..., ed−1}, {ed}) =

c(G′,{e1, e2, ..., ed−1}, {e′d}) and thus c(G′) = c(G).

Lemma 3.10. For s _{≥ 5 and m ≥ d + 3, if G = Ps}_{∪ H and G}′ = C3∪ Ps−3∪ H,

then c(G′) < c(G).

Proof. G = Ps ∪ H. Let v1, v2, ..., vs, ..., vm be an ordering of vertices of V (G)

such that Ps = v1v2...vs. Assume that E(G) = {e1, e2, ..., ed} where ed = {v3, v4}. G′ = C3 ∪ Ps−3 ∪ H. Let ed′ = {v1, v3}. We assume that V (G′) = V (G) and E(G′) ={e′d} ∪ (E(G) \ {ed}). Note that in G′, C3 = v1v2v3 and Ps−3 = v4v5...vs.

consider the bijection f : V (G) _{→ V (G}′) with f (v1) = v4, f (v3) = v1, f (v4) = v3, f (x) = x whenever x̸= vi for i = 1, 3, 4. See Figure 12 for easy reading. Then the mapping D_{7→ f(D) from} (V (G) d ) to (V (G′) d ) is injective.

Let D be a d-subset of V (G) that intersects every edge in_{e1, e2, ..., ed−1} but does

not intersect edge ed = {v3, v4}. We shall show that f(D) ⊆ V (G′) intersects every edge e_{∈ {e}1, e2, ..., ed−1} but does not intersect e′d ={v1, v3} in G′. Since v3, v4 ∈ D,/ v1 = f (v3), v3 = f (v4) /∈ f(D) and thus f(D) does not intersect e′d. We consider the

following three cases.

Case 1: e = _{v1, v2} or e = {v2, v3} in G′. Since v2 ∈ D, v2 = f (v2) ∈ f(D) and thus e is intersected by f (D).

Case 2: e = _{v4, v5} in G′. Since v5 ∈ D, f(v5) = v5 ∈ f(D) and thus e is intersected by f (D).

Case 3: e =_{vj, vk} with 5 ≤ j < k in G′. Since by assumption we have vj ∈ D

or vk ∈ D, we derive that vj = f (vj) ∈ f(D) or vk = f (vk) ∈ f(D), so f(D) does

intersect e =_{vj, vk} in G′.

So f (D)_{⊆ V (G}′) intersects every edge e∈ {e1, e2, ..., ed−1} but does not intersect e′_d={v1, v3} in G′. Hence c(G,{e1, e2, ..., ed−1}, {ed}) ≤ c(G′,{e1, e2, ..., ed−1}, {e′d}).

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Let D′ = K∪ {v2, v4, v6, v7, ..., vs} where K is a cover of H of size d − s + 3. Then it is clear that D′ intersects every edge in _{e1, e2, ..., ed−1} but does not intersect e′_d = {v1, v3} in G′. H has exactly m− s vertices. Since m ≥ d + 3, H has at least d− s + 3 vertices. Since H has exactly d − s + 1 edges, arbitrarily choosing one vertex

from each e _{∈ E(H) yields a cover of H of size at most d − s + 1. Thus H has a} cover of size d_{− s + 3. Since v}3, v5 ∈ D/ ′, the edge {v4, v5} is not covered by f−1(D′) in G. So c(G,_{e1, e2, ..., ed−1}, {ed}) < c(G′,{e1, e2, ..., ed−1}, {e′d}). By Lemma 3.2,

c(G′) < c(G).

3.2 Results

Given a set A, define A to be the complement of A. First, we give the following Theorem for common use.

Theorem 3.11. If m_{− d = 1, then M(m, m, d) is completely (d; m)-disjunct;}

more-over, M (m, m, d) is completely (s; d + 1)-disjunct for all 1 _{≤ s ≤} (2m)!

2m_m! − 1. If

m− d = 2, then M(m, m, d) is completely (d; z)-disjunct where z =(m₂)− d.

Proof. For any C0, C1, ..., Cd, let Ai = {D ⊆ C0|D ⊆ Ci,|D| = d} 1 ≤ i ≤ d

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c(G[C0; E1, ..., Ed]), is equal to|A1∩A2∩...∩Ad| = |S|− ∑ 1≤i≤d|Ai|+ ∑ 1≤i,j≤d|Ai∩ Aj|− ∑

1≤i,j,k≤d|Ai∩ Aj ∩ Ak| + ... + (−1) d_|A 1∩ A2∩ ... ∩ Ad| = (_m d ) − d(m−2 d ) . Case 1: when m_{− d = 2,} (m d ) − d(m−2 d ) =(_mm₋₂)− d(_mm−2₋₂)=(m₂)− d. Case 2: when m_{− d = 1,} (m d ) − d(m−2 d ) =(_mm₋₁)− d(_mm−2₋₁)=(m₁)= m.

Moreover, if m_{− d = 1, given any s + 1 columns C}0, C1, ..., Cs of M (m, m, d) with one

column C0designated, the number of error-correcting rows produced by (C0,{C1, ..., Cs}) is exactly the number of d-matchings D such that D _{⊆ C}0 and D * Ci for all i∈ [s].

Those d-matchings D such that D _{⊆ C}0 is exactly those d-subsets of C0, there are

d + 1 such D. Since |C0\ Ci| ≥ 2, all these d-matchings D such that D ⊆ C0 also

satisfies D _{* C}i for all i∈ [s]. So there are d + 1 error-correcting rows for any s + 1

columns C0, C1, ..., Cs and hence M (m, m, d) is completely (s; d + 1)-disjunct. Note

that by Theorem 1.2 there are (2m)!

2m_m! columns in M (m, m, d), so 1≤ s ≤

(2m)!

2m_m!− 1.

Second, we consider that d≤ ⌊m/2⌋. We have the following results.

Lemma 3.12. For d _{≤ ⌊m/2⌋, (1) if G contains only trees and not all P}2, then G

has at least one isolated vertex; (2) if G contains a component H that is not acyclic, then G has at least two isolated vertices.

Proof. Observe that we have the following three properties. Every tree except P1

has a property that the number of edges is equal to or greater than a half number of vertices. Every tree except P1 and P2 has a property that the number of edges is greater than a half number of vertices. Every component H that is not acyclic has a property that the number of edges is equal to or greater than the number of vertices, and H_{∪ P}1 has a property that the number of edges is greater than a half number of vertices.

(1) Suppose that G contains only trees not all P2 and G has no isolated point. Then summing up all components we have d >_{⌊m/2⌋, a contradiction.}

(2) Suppose that G contains a component H that is not acyclic and G has at most one isolated vertex. Then summing up all components we have d > _{⌊m/2⌋, a}

contradiction.

Lemma 3.13. For m_{≥ d + 3 and d ≤ ⌊m/2⌋, every optimal pattern contains P}2 as

a component.

Proof. If G contains a component that is not acyclic, then by Lemma 3.12 (2) G

has at least two isolated vertices, thus G is not optimal by Lemma 3.5. Thus for

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contrary that G does not contain P2 as a component. By Lemma 3.12 (1), G has at least one isolated vertex. Further, since d _{̸= 0, G = T ∪ P}1 ∪ H, where T is a tree other than P1 and P2. By Lemma 3.7, T is not a path and thus by Lemma 3.6, G is

not optimal, a contradiction.

Theorem 3.14. If d_{≤ ⌊m/2⌋, then M(m, m, d) is completely (d; 2}d_)-disjunct.

Proof. First, we want to show that if G is an optimal pattern then c(G)_{≥ 2}d_{. We}

use mathematical induction on d. If d = 1, the pattern consists of one P2 and m− 2 isolated points is the only one optimal pattern, so c(G) _{≥ 2. If d = 2, G has two} possible patterns that may be optimal, one pattern G1 consists of one P3 and m− 3 isolated points, the other pattern G2 consists of two P2 and m− 4 isolated points. Since c(G1) = m and c(G2) = 4 and since d ≤ ⌊m/2⌋, we have m ≥ 2d = 4, so c(G1) ≥ c(G2). Thus, the pattern G consists of two P2 and m− 4 isolated points is the only one optimal pattern, and c(G)_{≥ 4. Suppose that when d = k where k ≥ 2} we have c(G)_{≥ 2}k_{. Then when d = k + 1, since d}_{≥ 3 and d ≤ ⌊m/2⌋, so m ≥ d + 3.}

By Lemma 3.13, an optimal pattern G contains P2 as a component. Denote the set of all d-covers of G by Cd(G), note that|Cd(G)| = c(G). Let V (P2) ={v1, v2} and let

G = P2∪H. Then a d-subset D ∈ Cd(G) if and only if D ={v1}∪P or D = {v2}∪P

or D =_{v1, v2} ∪ Q, where P ∈ Cd−1(H) and Q ∈ Cd−2(H). Note that the number of D such that D =_{v1} ∪ P is exactly c(G − v1− v2), similarly for {v2} ∪ P . Then c(G)≥ 2·c(G−v1−v2)≥ 2·2k = 2k+1. So if G is an optimal pattern then c(G)≥ 2d. Second, we prove that there exists a pattern G′ such that c(G′) = 2d_{. This pattern} G′ is chosen by a graph consisting of d P2’s and m− 2d isolated points.

By the above two arguments we complete the proof.

4 Concluding Remark

Given _{|V (G)| = m and |E(G)| = d, by Theorem 2.5, it is (d + 1)-disjunct, but by} Lemma 3.11, it is completely ((m 2 ) − d) = ((d+2 2 ) − d)-disjunct whenever m = d + 2,

we can find that (d+2 2

)

− d is asymptotically greater than d + 1, so Theorem 2.5 still

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4.1 Summary

Now, we consider that d >⌊m/2⌋. We have the following Conjecture. This Conjecture is derived by using Lemma 3.4 to Lemma 3.10 on a good deal of patterns. For example, see Figure 13, Figure 14 and Figure 15.

Conjecture 4.1. For m > d >⌊m/2⌋,

(1) if m + d≡ 0 (mod 3), then the pattern consisting of m − d P2’s and (2d− m)/3 C3’s is optimal.

(2) if m+d≡ 1 (mod 3), then the pattern consisting of m−d−1 P2’s and (2d−m+1)/3

C3’s and one isolated vertex is optimal.

(3) if m+d≡ 2 (mod 3), then the pattern consisting of m−d−1 P2’s and (2d−m−1)/3

C3’s and one P3 is optimal.

Then, we use the method of generating functions to find the number z which is possibly true in completely (d; z)-disjunct of M (m, m, d). Let FG(x) :=

∑

x|c|

where c runs through all vertex cover of G. Thus the coefficient of xd _{in F}

G(x) is

the number of d-covers of G. Then we find that FP1(x) = x + 1, FP2(x) = x

2_{+ 2x,} FP3(x) = x

3_{+ 3x}2_{+ x, F}

C3(x) = x

3_{+ 3x}2_{. We use [x}d_{]f (x) to denote the coefficient}

of xd _{in polynomial f (x).}

Conjecture 4.2. For m > d >⌊m/2⌋,

(1) if m + d ≡ 0 (mod 3), then M(m, m, d) is completely (d; z)-disjunct where

z = [xd]f (x), f (x) = (x3+ 3x2)(2d−m)/3· (x2+ 2x)m−d.

z = [xd]f (x), f (x) = (x3+ 3x2)(2d−m+1)/3· (x2+ 2x)m−d−1· (x + 1).

z = [xd]f (x), f (x) = (x3+ 3x2)(2d−m−1)/3· (x2+ 2x)m−d−1· (x3+ 3x2+ x).

Let M (m, m, d) be completely (d; z)-disjunct. We want to know what number z is. By Property 2.2, this Conjecture gives us an upper bound of z whenever d >⌊m/2⌋, no matter whether this Conjecture is true or not. By program work, we give a summary for the number z in some cases of (m, d) where 1≤ d < m ≤ 15. See Figure 16.

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Figure 13: For m = 15 and d = 12, we guess that the pattern consisting of three P2’s and three C3’s is optimal.

Figure 14: For m = 15 and d = 10, we guess that the pattern consisting of four P2’s and two C3’s and one P1 is optimal.

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Figure 16: Each cell shows the number z such that M (m, m, d) is completely (d; z)-disjunct. Bold font: by program work. Normal font: by Theorem. Shadowed area: by Conjecture.

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4.2 Analysis

Consider m + d ≡ 0 (mod 3). Let G be the pattern consisting of m − d P2’s and (2d− m)/3 C3’s. To compute c(G), suppose that we choose two vertices in b C3’s and we choose three vertices in (2d− m)/3 − b C3’s. Then we must choose one vertex in m− d − b P2’s and two vertices in b P2’s to form a d-cover. So,

c(G) = min((2d−m)/3,m−d)_∑ b=0 ( (2d− m)/3 b )( m− d b ) (3)b(2)m−d−b ≤ (2d_∑−m)/3 b=0 ( (2d− m)/3 b ) (3)b· m∑−d b=0 ( m− d b ) (2)m−d−b = (2d_∑−m)/3 b=0 ( (2d− m)/3 b ) (3)b(1)(2d−m)/3−b· m∑−d b=0 ( m− d b ) (2)m−d−b(1)b = 4(2d−m)/3· 3m−d.

Since the number z of (d; z)-disjunct of M (m, m, d) is equal to or less than c(G), we have z ≤ 4(2d−m)/3· 3m−d _{whenever m + d}≡ 0 (mod 3).

Besides, let t = min((2d− m)/3, m − d), and note that (n k)

k≤ (n k

)

for latter use. If Conjecture 4.2 is true, then by above argument,

z = min((2d−m)/3,m−d)_∑ b=0 ( (2d− m)/3 b )( m− d b ) (3)b(2)m−d−b ≥ ( (2d− m)/3 t )( m− d t ) (3)t(2)m−d−t ≥ ((2d− m)/3 t ) t_{· (}m− d t ) t_{· (2)}m−d_{· (3/2)}t

Consider m + d _{≡ 1 (mod 3). Let G be the pattern consisting of m − d − 1 P}2’s and (2d_{− m + 1)/3 C}3’s and one isolated point. To compute c(G), we consider the following two cases.

Case 1, if the isolated vertex is selected. Suppose that we choose two vertices in

b C3’s and we choose three vertices in (2d− m + 1)/3 − b C3’s. Then we must choose one vertex in m_{− d − 1 − (b − 1) P}2’s and two vertices in b− 1 P2’s to form a d-cover. So in this case,

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c(G) = min((2d−m+1)/3,m−d)_∑ b=1 ( (2d− m + 1)/3 b )( m− d − 1 b− 1 ) (3)b(2)m−d−1−(b−1) ≤ (2d−m+1)/3_∑ b=0 ( (2d− m + 1)/3 b ) (3)b · m−d ∑ b=1 ( m− d − 1 b− 1 ) (2)m−d−1−(b−1) = (2d−m+1)/3_∑ b=0 ( (2d− m + 1)/3 b ) (3)b(1)(2d−m+1)/3−b· m−d ∑ b=1 ( m− d − 1 b− 1 ) (2)m−d−1−(b−1)(1)b−1 = 4(2d−m+1)/3· 3m−d−1.

Case 2, if the isolated vertex is not selected. Suppose that we choose two vertices in b C3’s and we choose three vertices in (2d− m + 1)/3 − b C3’s. Then we must choose one vertex in m_{− d − 1 − b P}2’s and two vertices in b P2’s to form a d-cover. So in this case, c(G) = min((2d−m+1)/3,m−d−1)_∑ b=0 ( (2d− m + 1)/3 b )( m− d − 1 b ) (3)b(2)m−d−1−b ≤ (2d−m+1)/3_∑ b=0 ( (2d− m + 1)/3 b ) (3)b · m∑−d−1 b=0 ( m− d − 1 b ) (2)m−d−1−b = (2d−m+1)/3_∑ b=0 ( (2d− m + 1)/3 b ) (3)b(1)(2d−m+1)/3−b· m∑−d−1 b=0 ( m− d − 1 b ) (2)m−d−1−b(1)b = 4(2d−m+1)/3· 3m−d−1.

Since the number z of (d; z)-disjunct of M (m, m, d) is equal to or less than c(G), we have z _{≤ 2 · 4}(2d−m+1)/3_{· 3}m−d−1 _{whenever m + d} _{≡ 1 (mod 3).}

Besides, let t = min((2d_{− m + 1)/3, m − d − 1), and note that (}n k)

k _≤ (n k

) for latter use. If Conjecture 4.2 is true, then by above argument,