邊著色圖中的混色子圖

國立交通大學

應用數學系

博 士 論 文

邊著色圖中的混色子圖

Multicolored Subgraphs in an

Edge-colored Graphs

博 士 生:羅元勳

指導教授:傅恆霖 教授

中 華 民 國 九 十 九 年 七 月

邊著色圖中的混色子圖

Multicolored Subgraphs in an

Edge-colored Graphs

博 士 生：羅元勳

Student:Yuan-Hsun Lo

指導教授：傅恆霖 Advisor:Hung-Lin Fu

國 立 交 通 大 學

應 用 數 學 系

博 士 論 文

A Dissertation

Submitted to Department of Applied Mathematics

College of Science

National Chiao Tung University

In Partial Fulfillment of the Requirements

For the Degree of

Doctor of Philosophy

In

Applied Mathematics

June 2010

Hsinchu, Taiwan, Republic of China

Abstract

A subgraph in an edge-colored graph is multicolored if all its edges receive distinct colors. In this dissertation, we first prove that a complete graph of order 2m (m= 2) can be properly edge-colored with 2m− 1 colors in such a way that the edges of K_2m can be partitioned into m isomorphic multicolored spanning trees. Then, for the complete graph on 2m + 1 vertices, we give a proper edge-coloring with 2m + 1 colors such that the edges of K_2m+1 can be partitioned into m multicolored Hamiltonian cycles.

In the second part, we first prove that if K_2m admits a proper (2m−1)-edge-coloring such that any two colors induce a 2-factor with each component a 4-cycle, then K_2m can be partitioned into m isomorphic multicolored spanning trees. As a consequence, we show the existence of three isomorphic multicolored spanning trees whenever m≥ 14. As to the complete graph of odd order, two multicolored isomorphic unicyclic spanning subgraphs can be found in an arbitrary proper (2m+1)-edge-coloring of K_2m+1.

If we drop the condition “isomorphic”, we prove that there exist Ω(√m) mutually

edge-disjoint multicolored spanning trees in any proper (2m−1)-edge-colored K_2m by applying a recursive construction. Using an analogous strategy, we can also find Ω(√m)

mutually edge-disjoint multicolored unicyclic spanning subgraphs in any proper (2m −1)-edge-colored K_2m−1.

Finally, we consider the problem of how to forbid a specific multicolored subgraph in a properly edge-colored complete bipartite graph. We (1) prove that for any integer k ≥ 2, if n≥ 5k − 6, then any properly n-edge-colored K_k,n contains a multicolored C_2k, and (2) determine the order of the properly edge-colored complete bipartite graphs which forbid multicolored 6-cycles.

摘要

在一個邊著色的圖中（以下的邊著色須滿足相接的兩條邊必為不

同顏色）

，如果有一個子圖它每個邊的顏色皆不相同，則稱這種子圖

為一個混色圖。在這篇論文中，首先我們證明點數為

的完全圖

(

)，存在一個

個顏色的邊著色，可以將

2m

2

m

≠

2m

−1

K

分解成

個互

相同構的混色懸掛樹。而對點數為 2

m

1

m

+ 的完全圖，我們也證明其邊

適當地著

2m

+1

個顏色後，

K

將可分解成 個混色的哈米爾頓圈。

第二部分，我們證明對於

個點的完全圖，如果有一種

個

顏色的邊著色使得任兩種顏色均會形成一組

的分割，則這種著色

的完全圖也可以分解成

個互相同構的混色懸掛樹。由這個結果，我

們可以證明在

2m

2

m

−

1

C

m

K

中(

m

≥

14

)，任意給定一種 2

m

− 個顏色的邊著

1

色，一定會存在三個同構的混色懸掛樹。至於對於點數為

的完

全圖，在任意的

個顏色邊著色下，也一定存在兩個同構的混色

子圖，其中這兩個子圖是懸掛單圈圖。

2

m

+

1

2

m

+1

若捨棄掉「同構」這個限制，我們利用一種遞迴的建構方法，可

以證明出在

K

中，任意給定一種 2m 1

− 個顏色的邊著色，存在約

( m

Ω

)

1

個邊互斥的混色懸掛樹。利用相同的策略－遞迴建構法，在

中，任意給定一種 2

K

m

− 個顏色的邊著色，我們也可找出約

( m

Ω

)

個邊互斥的混色懸掛單圈圖。

iii

6

最後，我們討論如何在一個邊著色的完全二部圖中避免某些特定

的混色子圖的出現。我們的貢獻有下列兩部分： (1) 對任意的

，

如果

，則任意 著色的完全二部圖

2

k

≥

5

n

≥

k

−

n

K

中一定找得到混色的

； (2) 刻劃出所有可避免混色

的完全二部圖。

C

C

Acknowledgement

“I hate the word ‘potential’—potential means you haven’t

gotten it done.”—Alex Rodriguez

這是

MLB 球星 A-Rod 在棒球場上說過的一句話，在做學問及成

長的道路上，何嘗不是如此。學生生涯已結束，面對接下來的挑戰，

我想用這句話來勉勵自己、鞭策自己，將「潛力」這兩字拋諸於腦後，

努力完成每件可能做到的事。

來到新竹恰好十年，在這條學習的路上，首先最感謝的是我的指

導教授－傅恆霖老師。從研究所到博士班，謝謝有老師您的不厭其煩

的教誨及包容，讓我能順利的突破重重的難關。尤其在最後這一年，

為了讓我能順利畢業，不辭辛勞地幫我驗證和修改期刊及博士論文，

一次又一次，感動的心真的不知如何描述。除了在課業上的幫忙，您

對於做人處世的態度和堅持也是讓我收獲良多。而在運動方面，也永

遠是我最敬重的教練。

除了傅老師，我也要感謝系上的翁志文老師、陳秋媛老師以及黃

大原老師，從您們身上我吸取到更多元的知識。特別是每學期的離散

專題，每個老師都有其堅持的地方，細嚼慢嚥，從中讓我體驗出做研

究真的是一件很有趣的事。另外，當然不能忘了系辦的漂亮三姐妹：

陳小姐、大張、小張。妳們就像我的大姊姊一樣，每次到系辦找妳們

v

聊天哈拉講八掛，都會覺得心情變得特別好。真的很謝謝妳們這些年

來的照顧，永遠不會忘記當年喊出的「一日工讀生，終生工讀生」。

同是傅家子弟的志弘、嘉芬、明輝、志銘、賓賓、robin、惠蘭、

裴、敏筠、智懷，研究的路上多虧了你們的幫忙，讓我收獲良多，得

以完成我的博士論文；特別是賓賓學長，除了在研究方面是我的榜樣

外，他也常提供很多寶貴的個人經驗給我參考，不謹改變我很多生活

態度，也開闊了我對數學的視野。另外，小培、啟賢是我生活上的好

朋友；小巴、川和和我是

AM93 留下攻讀博班的同學，有了你們的相

陪，讓我不會那麼孤單。

接下來要感謝的是系排的戰友們，在交大打滾了十年，排球一直

是我生活的重心之一。帶我進入排球世界的

robin、cool、god，國家

級教練傅老師，早期隊友致仁、拳頭、kert、劉砰砰、許老、明淇、

阿立，中期的小傑、小馬、大樹、小假死、理理人、蛤仔，一直到最

近的瑞毅、軒軒、企鵝、亥派、阿陰、小育等等，一起南爭北討、揮

撒汗水，嬴得冠軍的喜悅、輸掉比賽的眼淚，每一個珍貴的畫面，都

烙印在我心中。再來要對這些年來的同學、好麻吉們說聲感謝：拳頭、

小偉、樂咖、假死、小龍、小哈，有了你們相挺，讓我這十年過得更

加精采！

最後，要感謝的是我親愛的家人和芃瑀，謝謝你們的支持，讓我

可以堅持到最後；謝謝你們的體諒，讓我可以不顧一切地勇往直前。

最後的最後，謹用此論文獻給你們，表達我的感恩之心。

Contents

Abstract (in English) . . . i

Abstract (in Chinese) . . . ii

Acknowledgement . . . iv

Contents . . . vi

List of Figures . . . viii

List of Tables . . . x

1 Introduction and Preliminaries 1 1.1 Motivation . . . 1 1.2 Graph Terms . . . 2 1.3 Edge-coloring . . . 5 1.4 Basic Algebra . . . 8 1.5 Latin Square . . . 9 1.6 Parallelism Concept . . . 11 1.7 Known Results . . . 12

2 Multicolored Tree Parallelism 16 2.1 Known Results . . . 16

2.2 Main Results . . . 17

3 Multicolored Hamiltonian Cycle Parallelism 20 3.1 Known Results . . . 20

4 Multicolored Spanning Trees in Edge-Colored Complete Graphs 30

4.1 Isomorphic Multicolored Spanning Trees . . . 30

4.1.1 MST for C₄-factor edge-colored K_2m . . . 30

4.1.2 Main Results . . . 34

4.2 Multicolored Spanning Trees . . . 36

4.2.1 Recursive Construction . . . 36

4.2.2 Main Results . . . 45

5 Multicolored Unicyclic Spanning Subgraphs in Edge-Colored Complete Graphs 48 5.1 Isomorphic Multicolored Unicyclic Spanning Subgraphs . . . 48

5.2 Multicolored Unicyclic Spanning Subgraphs . . . 51

6 Forbidden Multicolored Cycles 53 6.1 Multicolored Subgraphs in Edge-colored Complete Graphs . . . 53

6.1.1 Multicolored Spanning Tree . . . 54

6.1.2 Multicolored Path . . . 54

6.1.3 Multicolored Cycle . . . 55

6.2 Forbidding Multicolored Cycles in Edge-colored Complete Bipartite Graphs 57 6.2.1 Forbidding Multicolored 2k-cycles . . . . 57

6.2.2 Determining F M C(6) . . . . 60

7 Conclusion and Remarks 65

List of Figures

1.1 Degree, neighborhood and regular . . . 3

1.2 spanning, factor and matching . . . 4

1.3 Hamiltonian cycle, tree and star . . . 4

1.4 Complete graph, complete bipartite and multipartite graph . . . 5

1.5 Two isomorphic graphs . . . 5

1.6 Three types of proper coloring . . . 6

1.7 ϕ−1 and vc notations . . . . 7

1.8 T and T [b, f ] . . . 7

1.9 Mutually orthogonal latin squares of order 3 and 4 . . . 10

1.10 2-group latin square of order 4 . . . 10

1.11 Idempotent commutative LS and corresponding edge-coloring . . . 11

2.1 K₆ admits an MTP. . . 17

2.2 K₁₂ admits an MTP. . . 18

3.1 Circulant latin square of order 7 . . . 21

3.2 K₇ admits an MHCP. . . 22

3.3 Two multicolored Hamiltonian cycles in 9-edge-colored K₉ . . . 26

3.4 E(1)∪ 7D(1) in K₃₅. . . 29

3.5 E(2)∪ 7D(2) in K₃₅. . . 29

4.1 4 transversals in L2. . . 31

4.2 4 transversals in Lk+1 _{constructed from A} 0 and A1. . . 31

4.3 Two isomorphic spanning trees of Case 1. . . 34

4.4 Two types of T₁. . . 35

4.5 T₃. . . 36

4.6 A properly 27-edge-colored K₂₈. . . 40

4.7 Two edge-disjoint multicolored spanning trees. . . 41

4.8 Three edge-disjoint multicolored spanning trees. . . 42

4.9 Four edge-disjoint multicolored spanning trees. . . 43

4.10 Estimate |U_n| from |U_n−1|. . . 47

5.1 Symmetric 7-total-coloring of K₇. . . 49

5.2 Three multicolored Hamiltonian cycles in symmetric 7-total-colored K₇. . . 49

5.3 (Case 1) Two multicolored isomorphic unicyclic subgraphs. . . 50

5.4 (Case 2) Two multicolored isomorphic unicyclic subgraphs. . . 51

6.1 The direct product of L and M . . . . 58

6.2 L₂ × M and the four copies of M . . . 59

6.3 L₂ × L₂ × L₂ . . . 61

6.4 A 3× 7 latin rectangle . . . 62

6.5 The 3× 7 latin rectangle . . . 63

List of Tables

1.1 The group table of Z₇, + . . . . 8

1.2 The group table of Z₇, + and Z∗₇,· . . . . 9

1.3 Three multicolored isomorphic spanning trees . . . 14

2.1 Color assignment of K₆ . . . 17

2.2 Color assignment of K₁₂ . . . 18

3.1 Color assignment of K₇ . . . 22

Chapter 1

Introduction and Preliminaries

1.1

Motivation

Graph decomposition and graph coloring are two of the most important topics in the study of graph theory. Graph decomposition deals with the partition of the edge set of a graph G into subsets each induces a graph in the list of prescribed subgraphs of G, and graph coloring studies the assignments of colors onto the vertex set of G or the edge set of

G or both or some well-understood areas. Either one of them has made a strong impact

to make graph theory more interesting and useful through the years.

The research on combining these two topics together starts at observing a subgraph in an edge-colored graph which has many colors. A subgraph whose edges are of distinct colors is known as a rainbow (or multicolored, heterochromatic) subgraph, see [36] for reference. In 1991, Alon, Brualdi and Shader [3] first showed that in any edge-coloring of K_n such that each color class forms a complete bipartite graph, there is a spanning tree of K_n with distinct colors. Some years later, in 1996, Brualdi and Hollingsworth [10] proved the existence of two disjoint multicolored spanning trees in any proper edge-coloring of K_2n. Then, they conjectured that a full partition into multicolored spanning trees is always possible. This conjecture encouraged many scholars to devote themselves to studying this kind of decomposition problem. In 2000, J. Krussel, S. Marshal and H. Verral [32] showed the existence of three edge-disjoint multicolored spanning trees about above conjecture, and it stopped for a while.

How about adding a condition that these spanning trees are isomorphic mutually? In 2002, G. M. Constantine [14] inserted a parallel concept into this problem. He proposed two conjectures. One of them is that any proper (2n− 1)-edge-coloring of K_2n allows a partition of the edges into multicolored isomorphic spanning trees. The other one is a weaker version of above by giving an edge-coloring ourselves and partitioning E(K_2n). Moreover, Constantine proved the latter conjecture on some specific orders.

It is not a coincidence that decomposing the complete graph with even order into spanning trees, because it is easy to decompose K_2ninto n Hamiltonian paths. Analogous to the complete graph of even order, how about that of odd order? Due to the chromatic index, it is natural to partition the graph into either unicyclic subgraphs or Hamiltonian cycles. In 2005, Constantine [15] partitioned K_2n+1into n multicolored Hamiltonian cycles by a given proper (2n + 1)-edge-coloring if n is a prime. Furthermore, he proposed a new conjecture that for any proper (2n + 1)-edge-coloring of K_2n+1, the edges can be partition into multicolored isomorphic spanning unicyclic subgraphs.

The above problems motivate us the study of this thesis.

1.2

Graph Terms

In this section, we first introduce the terminologies and definitions of graphs. For details, the readers may refer to the book “Introduction to Graph Theory” by D. B. West.[35]

A graph G is a triple consisting of a vertex set V (G), an edge set E(G), and a relation that associates each edge with two vertices called its endpoints. A loop is an edge whose endpoints are equal. Multiedges are edges having the same pair of endpoints. A simple

graph is a graph without loops and multiedges. In this thesis, all the graphs we consider

are simple. The size of the vertex set V (G), |V (G)|, is called the order of G, and the size of the edge set E(G), |E(G)|, is called the size of G.

We also say that u and v are adjacent to each other. For every v∈ V (G), N(v) denotes the neighborhood of v; that is, all vertices of N (v) are adjacent to v. The degree of v in a graph G, written d_G(v) or d(G), is the number of neighbors of v in G. The maximum degree is Δ(G), and the minimum degree is δ(G). Moreover, G is regular if Δ(G) = δ(G), and it is said to be k-regular if the common degree is k.

a f e d c b Ӕ= 4 Ӭ= 1 N(c)={a, d, f} 3-regular

Figure 1.1: Degree, neighborhood and regular

A path is a simple graph whose vertices can be ordered so that two vertices are adjacent if and only if they are consecutive in the list. A graph G is connected if each pair of vertices in G belongs to a path; otherwise, G is disconnected.

A subgraph of a graph G is a graph H such that V (H) ⊆ V (G) and E(H) ⊆ E(G) and the assignment of endpoints to edges in H is the same as in G. Given S be a subset of vertex set V (G), the induced subgraph determined by S, denoted byS_G, is a subgraph of G such that for any u, v∈ S, u is adjacent to v in S_G if u is adjacent to v in G.

A spanning subgraph (or factor) of G is a subgraph with vertex set V (G). A spanning subgraph is said to be k-factor if it is k-regular.

A matching of size k in G is a set of k pairwise disjoint edges. If a matching covers all vertices of G, then it is a perfect matching. Accordingly, a perfect matching and an 1-factor are almost the same thing. In Figure 1.2, the edge set {af, bg, ch, di, ej} is a perfect matching of G and it induces an 1-factor.

A cycle is a graph with an equal number of vertices and edges whose vertices can be placed around a circle so that two vertices are adjacent if and only if they appear consecutively along the circle. A cycle with n vertices is denoted by C_n. A Hamiltonian

G a j i h g f e d c b spanning subgraph of G 1-factor 2-factor

Figure 1.2: spanning, factor and matching

graph is a graph with a spanning cycle, also called a Hamiltonian cycle. A graph with

exactly one cycle is unicyclic; therefore, a hamiltonian cycle in a hamiltonian graph is a unicyclic subgraph.

In contrast, a graph with no cycle is acyclic. A tree is a connected acyclic graph. A

leaf (or pendant vertex) in a tree is a vertex of degree 1. A star is a tree consisting of

one vertex adjacent to all the others, and the particular vertex is said to be the root (or

center) of the star. Let S_x denote a star with center x.

G a e f d c b

Hamiltonian cycle Tree Sf

f

Figure 1.3: Hamiltonian cycle, tree and star

A clique in a graph is a set of pairwise adjacent vertices. An independent set in a graph is a set of pairwise nonadjacent vertices.

A complete graph is a simple graph whose vertices are pairwise adjacent, and the complete graph with n vertices is denoted by K_n. A graph G is bipartite if V (G) is the union of two disjoint independent sets, called partite sets of G. A graph G is

m-partite if V (G) can be expressed as the union of m independent sets. A complete bim-partite graph is a bipartite graph such that two vertices are adjacent if and only if they are in

different partite sets. When the sets have the sizes s and t, the complete bipartite graph is denoted by K_s,t. If the sets have the same size n, the complete bipartite graph is said to be balanced, denoted by K_n,n. Similarly, the complete m-partite graph is denoted by

K_s1,s2,...,sm where s_i is the size of the i-th partite set, and the balanced complete m-partite graph is denoted by K_m(n) where each partite set has n vertices.

K5 K2,4 K2,2,2,2(or K4(2))

Figure 1.4: Complete graph, complete bipartite and multipartite graph

An isomorphism from a graph G to a graph H is a bijection f : V (G)→ V (H) such that uv∈ E(G) if and only if f(u)f(v) ∈ E(H). We say “G is isomorphic to H”, written

G ∼= H, if there is an isomorphism from G to H.

x1 x10 x9 x8 x7 x6 x5 x4 x3 x2 x1 x5 x6 x2 x9 x8 x7 x4 x3 x10

Figure 1.5: Two isomorphic graphs

1.3

Edge-coloring

A k-coloring of a graph G is a mapping from V (G) into a set of colors {1, 2, . . . , k}, referred as a color set. The vertices of one color form a color class. A k-coloring is proper if adjacent vertices have different colors. A graph is k-colorable if it has a proper k-coloring; furthermore, name the least k such that G is k-colorable be the chromatic number of G, written χ(G).

Analogous to k-coloring, a k-edge-coloring, proper k-edge-coloring and k-edge-colorable can be defined by replacing V (G) with E(G), and let the chromatic index χ(G) be the least k such that G is k-edge-colorable. Combining these two kinds of colorings, an (proper) k-total-coloring of a graph G is a mapping from V (G)∪ E(G) into a set of colors

{1, 2, . . . , k} such that (i) adjacent vertices in G receive distinct colors, (ii) incident edges

in G receive distinct colors, and (iii) any vertex and its incident edges receive distinct colors. 1 3 3 2 2 3 3 2 1 1 4 4 3 3 2 1 1 4 4 1 5 4 3 2

3-coloring 4-edge-coloring 5-total-coloring

Figure 1.6: Three types of proper coloring

Figure 1.6 shows the three types of proper coloring: (vertex-)coloring, edge-coloring and total-coloring. Note here we usually use Arabic numerals to denote the colors; how-ever, in same chapters we take symbols such as c₁, c₂, . . . or (0, 0), (0, 1), . . . to denote

colors. No matter what they are, different symbols indicate different colors. Here are some famous results about colorings, edge-colorings, and total-colorings.

Theorem 1.3.1. (Brooks [9]) If G is a connected graph other than a complete graph or

an odd cycle, then χ(G)≤ Δ(G).

Theorem 1.3.2. (Vizing [34]) If G is simple graph, then Δ(G)≤ χ(G)≤ Δ(G) + 1. Theorem 1.3.3. [37] If n is an odd positive integer, then K_n has an n-total-coloring.

According to Vizing’s theorem, for simple graphs, there are only two possibilities for χ. A simple graph G is of Class 1 if χ(G) = Δ(G), while it is of Class 2 if χ(G) = Δ(G)+1. It is not hard to check that K_2m is Class 1 and K_2m+1 is Class 2.

In this thesis, we mainly focus on proper edge-coloring. Let ϕ be a proper (2m −1)-edge-coloring of K_2m and C be the color set. For each x ∈ V (K_2m), define ϕ_x as the

mapping from V (K_2m)\ {x} to C by ϕ_x(y) = c if ϕ(xy) = c. Clearly, ϕ_x is bijective. For each vertex x, let ϕ−1_x (c) be the vertex adjacent to x with the edge colored c. For convenience, we use vc to denote the edge incident to v with color c.

a d c b 3 1 1 2 3 2 ӽ-1a(1)=b ӽ-1a(3)=c ӽ-1a(2)=d a 1 =abτυ a 2 =adτυ a 3 =acτυ

Figure 1.7: ϕ−1 and vc notations

A subgraph in an edge-colored graph is said to be multicolored (or rainbow,

heterochro-matic) if no two edges have the same color. Suppose T is a multicolored spanning tree of K_2m with two leaves x₁ and x₂. Let the edges in T incident to x₁ and x₂ be e₁ and e₂ respectively, and ϕ(e₁) = c₁, ϕ(e₂) = c₂. Then let T [x₁, x₂] be the tree obtained from T by removing the edges e₁, e₂ and adding the edges x₁c₂, x₂c₁.

f e d c b a 4 1 1 1 2 2 5 5 3 5 4 3 3 2 4 _f e d c b a 1 2 5 3 4 f e d c b a 5 3 4 ₁ 2

K

T

T[b , f ]

Figure 1.8 provides a properly 5-edge-colored K₆ and one of its multicolored spanning tree T . Given b and f be two leaves in T . It is easy to see that the tree T [b, f ] is still multicolored and spanning.

1.4

Basic Algebra

Definition 1.4.1. A group G, ∗ is a nonempty set G with a binary operation ∗ such that:

(1) a, b∈ G implies that a ∗ b ∈ G.

(2) For all a, b, c∈ G, we have a ∗ (b ∗ c) = (a ∗ b) ∗ c.

(3) There is an element e∈ G, say identity, such that a ∗ e = e ∗ a = a for any a ∈ G. (4) For every a ∈ G there exists an element b ∈ G such that a ∗ b = b ∗ a = e.

A group G, ∗ is said to be abelian if a ∗ b = b ∗ a for all a, b ∈ G. If the set G has an finite number of elements, we sayG, ∗ is a finite group.

For each positive integer n, we can partition Z+, all positive integers, into n subsets according to whenever the remainders of two positive integers divided by n is the same. These subsets are called the residue classes modulo n in Z+. If a and b have the same remainder divided by n, then we write a≡ b (mod n), read, ”a is congruent to b modulo

n.” For convenient, we use Z_n = {0, 1, 2, . . . , n − 1} to denote the set of residue classes modulo n. It is easy to see that Z_n, n ∈ Z+, is a finite group under the usual addition modulo n. Table 1.1 presents the structure of the group Z₇, +.

+ 0 1 2 3 4 5 6 0 0 1 2 3 4 5 6 1 1 2 3 4 5 6 0 2 2 3 4 5 6 0 1 3 3 4 5 6 0 1 2 4 4 5 6 0 1 2 3 5 5 6 0 1 2 3 4 6 6 0 1 2 3 4 5

Table 1.1: The group table of Z₇, +

Definition 1.4.2. A fieldF, +, · is a nonempty set F with two binary operations + and

(1) F, + is an abelian group with identity 0.

(2) F∗,· is an abelian group with identity 1, where F∗ = F \ {0}.

(3) For all a, b, c∈ F , we have a · (b + c) = a · b + a · c and (b + c) · a = b · a + c · a. Given a prime p, it is not hard to check that Z_p is a field under usual addition and multiplication modulo p. Table 1.2 presents the structure of the field Z₇, +,·.

+ 0 1 2 3 4 5 6 0 0 1 2 3 4 5 6 1 1 2 3 4 5 6 0 2 2 3 4 5 6 0 1 3 3 4 5 6 0 1 2 4 4 5 6 0 1 2 3 5 5 6 0 1 2 3 4 6 6 0 1 2 3 4 5 · 1 2 3 4 5 6 1 1 2 3 4 5 6 2 2 4 6 1 3 5 3 3 6 2 5 1 4 4 4 1 5 2 6 3 5 5 3 1 6 4 2 6 6 5 4 3 2 1

Table 1.2: The group table of Z₇, + and Z∗₇,·

The group Z_n, n ∈ Z+, and the filed Z_p, p∈ Z+ a prime, play two important roles in the description of proofs to our results. For more information about algebra, we refer to [19] and [27].

1.5

Latin Square

Let S be an n-set. A latin square of order n based on S is an n× n array such that each element of S occurs in each row and each column exactly once. For example, 0 1

1 0

is a latin square of order 2 based on{0, 1} = Z₂. Since this latin square corresponds to a group table ofZ₂, +, the latin square is also known as a 2-group latin square.

For convenience, we denote a latin square of order n based on S by L = [ l_i,j ] where

l_i,j ∈ S and i, j ∈ Z_n. Let L = [ l_i,j ] and M = [ m_i,j ] be two latin squares of order

n based on S. Then L = [ l_i,j ] and M = [ m_i,j ] are a pair of orthogonal latin squares, denoted by L⊥ M, if and only if {(l_i,j, m_i,j)| 1 ≤ i, j ≤ n} = S × S.

Let L = [ l_i,j ] and M = [ m_i,j ] be two latin squares of order l based on S and m based on T , respectively. Then the direct product of L and M , L× M = [ h_i,j ], is a latin

1 2 2 0 0 1 0 1 2 1 2 0 1 2 0 0 2 1 0 1 1 0 2 3 2 3 0 3 2 1 3 2 1 0 0 1 3 2 1 0 2 1 3 2 3 0 3 0 2 1 0 1 2 3 3 2 2 0 1 1 0 3 3 1 0 2

Figure 1.9: Mutually orthogonal latin squares of order 3 and 4

square of order l· m based on S × T , where h_x,y = ( l_a,b, m_c,d ) provided that x = ma + c and y = mb + d. For example, let L be the 2-group latin square, then L× L (or L2) is a latin square of order 4 based on Z₂× Z₂ as in Figure 1.10.

(0,1) (0,0) (1,0) (0,0) (0,0) (0,0) (0,1) (0,1) (0,1) (1,1) (1,0) (1,0) (1,0) (1,1) (1,1) (1,1) 0 1 2 3 0 1 2 3

Figure 1.10: 2-group latin square of order 4

A transversal of a latin square of order n is a set of n entries from each column and each row such that these n entries are all distinct. For example, in Figure 1.10,

{h0,0, h1,2, h2,3, h3,1} is a transversal. It is not difficult to see L × L does have 4 disjoint

transversals. Clearly, if a latin square of order n has n disjoint transversals, then it has an orthogonal latin square mate.

A latin square L = [l_i,j] is commutative if l_i,j = l_j,i for each pair of distinct i and j, and L is idempotent if l_i,i = i, i∈ [n]. Furthermore, L is circulant if l_i,j = l_i−1,j+1 where the indices i, j are taken modulo n.

a corresponding relationship between L and a properly n-edge-colored K_n. Let V (K_n) =

{v1, v2,· · · , vn} and the edge vivj is colored with li,j for each 1 ≤ i = j ≤ m, then we

have a proper n-edge-coloring of K_n, and vice versa.

1

4

2

5

3

2

5

3

1

4

3

1

4

2

5

5

3

1

4

2

4

2

5

3

1

v

v

v

v

v

Figure 1.11: Idempotent commutative LS and corresponding edge-coloring

A similar idea shows that a latin square of order n corresponds to an n-edge-coloring of the complete bipartite graph K_n,n. Let{u₁, u₂,· · · , u_n} and {v₁, v₂,· · · , v_n} be the two

partite sets of K_n,n and the edge u_iv_j be colored with l_i,j where L = [l_i,j] is a latin square , we have a proper n-edge-coloring of K_n,n. Therefore, a transversal of a latin square of order n is corresponded to a multicolored perfect matching in a properly n-edge-colored

K_n,n.

For more information on latin squares, we refer to [16].

1.6

Parallelism Concept

The notion of parallelism has always played an important role in mathematics. Euclid’s famous ”parallel postulate” asserted that, given any line and any point in the plane, the given point lies on a unique line parallel to the given line.

In a graph G = (V, E) we may consider each edge e as a set {x, y} consisting of the two vertices incident to e. Then, two edges e, e are called parallel (or independent in this case) if they are disjoint, i.e., e∩ e = φ. As an extension, two subgraphs are said to be parallel if they use no common edges. Furthermore, if all edges of a graph G can be covered by copies of a subgraph H, then we say the set of these copies is a parallelism of

H’s. Therefore, an 1-factorization can be considered as a parallelism of 1-factors.

We mainly consider two aspects of parallelism in a complete graphs. Firstly, given a proper χ(K_n)-edge-coloring of K_n. Then, the set of edges in a color class is parallel to another set of edges induced by a distinct color. Since each color class is a matching, a proper χ(K_n)-edge-coloring of K_n is a typical parallelism of matchings.

The second parallelism we will mention is parallelism of isomorphic spanning trees (respectively spanning unicyclic subgraphs) in a complete graph of even order (respectively odd order). Given a complete graph of even order and a partition of all edges into isomorphic spanning trees, it provides a parallelism of spanning trees. Furthermore, if the complete graph K_2m is properly (2m−1)-edge-colored and the edges of E(K_2m) can be decomposed into m isomorphic multicolored spanning trees, then we have a double

parallelism of isomorphic spanning trees, or parallelism of isomorphic spanning trees for

short. Subsequently, when it comes to a complete graph of odd order, we have a double parallelism of isomorphic spanning unicyclic subgraphs.

Harary [26] proposed several examples of a hierarchy of parallel structures in a graph in 1993. For more information about parallelism concept, see [11] for an introduction of a parallelism of complete designs. It is worth of mention here that the parallel concept plays important roles in applications. An application of parallelisms of complete designs to population genetics data can be found in [7]. Parallelisms are also useful in partitioning consecutive positive integers into sets of equal size with equal power sums [30]. In addition, the generating function of the multicolored spanning trees in any edge colored graph can be expressed as a sum of formal determinants, in [5] and [6]. These results have been used in constructing parallelisms of multicolored trees in complete graphs on a small number of vertices.

1.7

Known Results

Lemma 1.7.1. [35] ∀m ∈ N, χ(K_2m) = 2m− 1 and χ(K_2m+1) = 2m + 1.

Base on Lemma 1.7.1 and the fact that K_2m can be partitioned into paths, Brualdi and Hollingsworth first made the following conjecture in 1996.

Conjecture 1.7.2. [10] If K_2m is properly (2m−1)-edge-colored, then the edges of K_2m can be partitioned into m multicolored spanning trees except when m = 2.

Meanwhile, they also proved the following theorem.

Theorem 1.7.3. [10] If the complete graph K_2m, m > 2, is properly (2m−1)-edge-colored, then there exist two edge-disjoint multicolored spanning trees.

Krussel, Marshall and Verall [32] extend Theorem 1.7.3 to three multicolored spanning trees.

Theorem 1.7.4. [32] If m > 2, then in any proper edge-coloring of K_2m with 2m−1 colors, there exist three edge-disjoint multicolored spanning trees.

It will be more difficult if the desired multicolored spanning trees are mutually iso-morphic. Here is an example of a 5-edge-colored K₆.

Example 1.7.5. In K₆, let {x₁, x₂, x₃, x₄, x₅, x₆} be the vertex set and {c₁, c₂, . . . , c₅} be

the color set. The following table shows an assignment of colors and a partition of the edge set. The ith row denotes the edges which are colored with c_i for 1 ≤ i ≤ 5; and, the jth column denotes the edges contained in the jth multicolored spanning tree for 1≤ j ≤ 3.

It is not difficult to see that we have a double parallelism of isomorphic spanning trees of K₆. Formally, we say that the complete graph K_2m admits a multicolored tree

parallelism (MTP), if there exists a proper (2m−1)-edge-coloring of K_2m such that the edges can be partitioned into m isomorphic multicolored spanning trees. The following result shown by Constatine [14] provides an infinite number of complete graphs which admit MTP.

T₁ T₂ T₃ c₁ x₃x₅ x₄x₆ x₁x₂ c₂ x₂x₄ x₁x₅ x₃x₆ c₃ x₂x₅ x₃x₄ x₁x₆ c₄ x₂x₆ x₁x₃ x₄x₅ c₅ x₁x₄ x₂x₃ x₅x₆

Table 1.3: Three multicolored isomorphic spanning trees

Theorem 1.7.6. [14] The graph K_nadmits an MTP whenever n = 2k_{, k > 2, or n = 5·2}k_,

k≥ 1.

He also posed the following two conjectures.

Conjecture 1.7.7. (Weak version) [14] K_2m can be properly edge-colored with 2m− 1 colors in such a way that the edges can be partitioned into m multicolored isomorphic spanning trees whenever m > 2.

Conjecture 1.7.8. (Strong version) [14] If K_2m is properly (2m−1)-edge-colored, then the edges of K_2mcan be partitioned into m multicolored isomorphic spanning trees except when m = 2.

On the other direction, we can also consider the complete graph of odd order. Since

χ(K_2m+1) = 2m + 1, the maximal size of a multicolored subgraph of a properly (2m+1)-edge-colored K_2m+1 is 2m + 1. So, it is natural to ask if there also exists a partition of the edges of a properly (2m+1)-edge-colored K_2m+1 into multicolored subgraphs of size 2m + 1. Constatine gave the following result.

Theorem 1.7.9. [15] If n is an odd prime, then there exists a proper n-edge-coloring of

K_n such that the edges can be partitioned into multicolored Hamiltonian cycles.

In fact, Constantine proposed two conjectures relative to this topic.

Conjecture 1.7.10. (Weak version) [15] For any odd integer n≥ 3, there exists a proper

n-edge-coloring of K_nsuch that all edges can be partitioned into multicolored Hamiltonian cycles.

Conjecture 1.7.11. (Strong version) [15] Any proper coloring of the edges of a complete graph on an odd number of vertices allows a partition of the edges into multicolored isomorphic unicyclic subgraphs.

In addition, there are results relevant to the existence of a multicolored subgraph in an edge-colored graph. Here we list a couple of them.

Theorem 1.7.12. [36] For m ≥ 3, every properly (2m−1)-edge-colored K_2m has a mul-ticolored perfect matching.

Theorem 1.7.13. [28] For any proper n-edge-coloring in K_n,n, there exists a multicolored matching with size at least n− (11.053)log2n.

The rest of this thesis is organized as follows. In Chapter 2 and Chapter 3, we deal with the decomposition of properly edge-colored complete graphs (assigned colorings) of even and odd order into multicolored isomorphic spanning trees and multicolored Hamiltonian cycles, respectively. In the next two chapters, all colorings we consider are given. First, in Chapter 4, we prove the existence of three edge-disjoint multicolored isomorphic spanning trees in a properly (2m−1)-edge-colored K_2m whenever m≥ 14, and about√m− 1

edge-disjoint multicolored spanning trees in K_2m. In Chapter 5, we tackle the cases on K_2m+1. Finally, in Chapter 6, the forbidden type problem is concerned. Mainly, we determine the order of those properly edge-colored complete bipartite graphs which forbid multicolored

Chapter 2

Multicolored Tree Parallelism

2.1

Known Results

Definition 2.1.1. We say that the complete graph K_2m admits a multicolored tree par-allelism (MTP) if there exists a proper (2m−1)-edge-coloring of K_2m for which all edges can be partitioned into m isomorphic multicolored spanning trees.

It is clear that the complete graph K₄does not admit an MTP. We note here that such a partition of the edges of K_2mcan be viewed as a double parallelism of K_2mas defined in Section 1.6. In fact, finding a partition as obtained above corresponds to an arrangement of the edges of K_2m into an array of 2m− 1 rows and m columns such that each row contains the edges with the same color which form a perfect matching and the edges in each column form a multicolored spanning tree of K_2m; moreover, all the m spanning trees are isomorphic.

Example 2.1.2. The complete graph K₆ admits an MTP. To see this, consider the complete graph K₆ with the vertex set {x₁, x₂, x₃, x₄, x₅, x₆}. Table 2.1 gives a proper

edge-coloring of K₆ with the colors c₁, c₂, c₃, c₄, c₅ as well as an MTP for it. The ith row of this table is the set of all edges with color c_i. Each column denotes the edges of a multicolored spanning tree. Figure 2.1 shows that the spanning trees T₁, T₂, T₃ are isomorphic.

T₁ T₂ T₃ c₁ x₃x₅ x₄x₆ x₁x₂ c₂ x₂x₄ x₁x₅ x₃x₆ c₃ x₂x₅ x₃x₄ x₁x₆ c₄ x₂x₆ x₁x₃ x₄x₅ c₅ x₁x₄ x₂x₃ x₅x₆

Table 2.1: Color assignment of K₆

T1 x2 T2 T3 x3 x6 x4 x5 x1 x3 x5 x2 x4 x1 x6 x6 x4 x3 x1 x5 x2 Figure 2.1: K₆ admits an MTP.

The following result has been proved in [14].

Theorem 2.1.3. [14] If m= 1, 3 and K_2m admits an MTP, then K₂r_m admits an MTP,

for all r≥ 1.

The mail goal of this chapter is to prove Conjecture 1.7.7, which states that K_2m admits an MTP for m > 2.

2.2

Main Results

P. Cameron [11] found a decomposition of K_6,6into six isomorphic multicolored graphs

K_1,3 ∪ 3K₂ ∪ 2K₁ by using the software Gap. In the next lemma, we use Cameron’s decomposition to find an MTP for K₁₂.

Lemma 2.2.1. The complete graph K₁₂ admits an MTP.

Proof. Consider the complete graph K₁₂ with the vertex set {u₁, . . . , u₆, v₁, . . . , v₆}.

Table 2.2 gives a proper edge coloring of K₁₂with colors c₁, . . . , c₁₁ as well as an MTP for it. The ith row of this table is the set of all edges with color c_i. Each column denotes the

edges of a multicolored spanning tree. Note that the first six rows of the table determine a decomposition of K_6,6 into six multicolored subgraphs to K_1,3∪ 3K₂∪ 2K₁.

T₁ T₂ T₃ T₄ T₅ T₆ c₁ u₂v₅ u₁v₆ u₆v₁ u₃v₂ u₄v₃ u₅v₄ c₂ u₂v₃ u₅v₂ u₆v₆ u₄v₅ u₃v₄ u₁v₁ c₃ u₄v₁ u₃v₃ u₆v₄ u₁v₂ u₅v₅ u₂v₆ c₄ u₁v₄ u₃v₅ u₅v₃ u₆v₂ u₂v₁ u₄v₆ c₅ u₂v₂ u₄v₄ u₁v₅ u₅v₁ u₆v₃ u₃v₆ c₆ u₅v₆ u₃v₁ u₄v₂ u₂v₄ u₁v₃ u₆v₅ c₇ u₃u₅ u₄u₆ u₁u₂ v₃v₅ v₄v₆ v₁v₂ c₈ u₂u₄ u₁u₅ u₃u₆ v₂v₄ v₁v₅ v₃v₆ c₉ u₂u₅ u₃u₄ u₁u₆ v₂v₅ v₃v₄ v₁v₆ c₁₀ u₂u₆ u₁u₃ u₄u₅ v₂v₆ v₁v₃ v₄v₅ c₁₁ u₁u₄ u₂u₃ u₅u₆ v₁v₄ v₂v₃ v₅v₆

Table 2.2: Color assignment of K₁₂

T1 u2 T2 T3 u5 u4 u6 u3 u1 v2 v3 v5 v6 v4 v1 u3 u4 u1 u2 u6 u5 v1 v3 v5 v4 v4 v6 u6 u1 u5 u3 u2 u4 v1 v4 v6 v5 v2 v3 T4 v2 T5 T6 v5 v4 v6 v3 v1 u1 u3 u6 u4 u5 u2 v3 v4 v1 v2 v6 v5 u1 u4 u6 u3 u5 u2 v6 v1 v5 v3 v2 v4 u2 u3 u4 u1 u5 u6 Figure 2.2: K₁₂ admits an MTP.

Now, we are ready to prove our main result.

Theorem 2.2.2. For m= 2, K_2m admits an MTP.

Proof. First, suppose that m is an odd integer. Consider the complete graph K_2m defined on the set A∪ B where A = {a₁, . . . , a_m} and B = {b₁, . . . b_m}. For convenience,

let G and H be the complete graphs on the sets A and B, respectively. Since m is odd, G has a total coloring π which uses m colors, 1, . . . , m. Now, define a proper edge-coloring

ϕ of K_2m as follows:

(a) For each edge a_ja_k ∈ E(G), let ϕ(a_ja_k) = π(a_ja_k); (b) For each edge b_jb_k ∈ E(H), let ϕ(b_jb_k) = π(a_ja_k); (c) For each edge a_ib_i, 1 ≤ i ≤ m, let ϕ(a_ib_i) = π(a_i); and

(d) For each edge a_jb_k, j = k, let ϕ(a_jb_k) = m + t where t ≡ k − j (mod m) and

t∈ {1, 2, · · · , m − 1}.

Clearly, ϕ is a proper (2m−1)-edge-coloring of K_2m. It is left to decompose K_2m into

m multicolored isomorphic spanning trees. First, for each i ∈ {1, 2, 3, · · · , m}, let T_i be defined on the set A∪ B and E(T_i) = {a_ia_{i+2t (mod m)}, b_ib_{i+2t−1 (mod m)}, b_ia_{i+2t−1 (mod m)}, a_i+1b_{i+2t (mod m)} | t = 1, 2, · · · , m−1₂ } ∪ {a_ib_i}. Then, it is easy to check that each T_i is a multicolored spanning tree of K_2m, and all the T_i’s are isomorphic.

Now, if m is not an odd integer, then 2m = 2t_{· m} _{where t ≥ 2 and m} _{is odd. In}

case where m = 1, t must be at least 3. Then it is direct consequence of Theorem 1.7.6. Assume m ≥ 3. Thus, K₂t_m admits an MTP by Theorem 2.1.3 except when m = 3 and

t = 2. Since this case can be handled by Lemma 2.2.1, we conclude the proof.

We note here that the above theorem proves Conjecture 1.7.7 and the result has been included in a paper written jointly with S. Akbari, A. Alipour and H. L. Fu [2].

Chapter 3

Multicolored Hamiltonian Cycle

Parallelism

To extend the study in Chapter 2 of parallelism to the other graph, K_2m+1 deserves to be considered first. Since χ(K_2m+1) = 2m + 1, the multicolored subgraph we consider has 2m + 1 edges. Thus, a multicolored Hamiltonian cycle in K_2m+1 is the best candidate for the subgraphs. In this chapter, we shall prove that for each positive integer m, there exists a proper (2m+1)-edge-coloring of K_2m+1for which all edges can be partitioned into multicolored Hamiltonian cycles. Obviously, any two Hamiltonian cycles are isomorphic and therefore we have another parallelism if exists.

3.1

Known Results

Definition 3.1.1. We say that the complete graph K_2m+1 admits a multicolored Hamil-tonian cycle parallelism (MHCP) if there exists a proper (2m+1)-edge-coloring of K_2m+1 for which all edges can be partitioned into m multicolored Hamiltonian cycles.

Review that a latin square L = [_i,j] is commutative if _i,j = _j,ifor each pair of distinct

i and j inZ_n, and L is idempotent if _i,i = i for i∈ Z_n. It is well-known that an idempotent commutative latin square of order n exists if and only if n is odd. For the convenience in the proof of our main result, we shall use a special latin square M = [m_i,j] of odd order

n which is a circulant latin square with 1st row (0,n+1

2 , 1,n+32 , 2,· · · ,n+n−22 ,n−12 ). Figure

Figure 3.1: Circulant latin square of order 7

A similar idea shows that a latin square of order n corresponds to a proper n-edge-coloring of the complete bipartite graph K_n,n. Let{u₀, u₁,· · · , u_n−1} and {v₀, v₁,· · · , v_n−1}

be the two partite sets of K_n,nand let M = [m_i,j] be a circulant latin square of order n with the first row as described in the preceding paragraph. Color edge u_iv_j of K_n,n with color

m_i,j and observe that the result is a proper n-edge-coloring of K_n,n with the extra prop-erty that for 0≤ j ≤ n − 1, the perfect matching {u₀v_j, u₁v_j+1, u₂v_j+2,· · · , u_n−1v_j+n−1},

where the indices of v_i are taken modulo n with i ∈ Z_n, is multicolored. We note here that if we permute the entries of M , we obtain another proper n-edge-coloring of K_n,n which has the same property as above.

The following result by Constantine appears in [15].

Theorem 3.1.2. [15] If n is an odd prime, then there exists a proper n-edge-coloring of

K_n such that all edges can be partitioned into multicolored Hamiltonian cycles.

Note that this result can be obtained by using a circulant latin square of order n to color the edges of K_n and the Hamiltonian cycles are corresponding to 1st, 2nd, · · · , (n−1₂ )-th sub-diagonals respectively.

Example 3.1.3. In K₇, the edges are colored by using Figure 3.1, and the three cycles are induced by {x₀x_i+1, x₁x_i+2,· · · , x₆x_i} where V (K₇) = {x₀, x₁,· · · , x₆}, i = 0, 1, 2, where

the sub-indices are in [n]. See Table 3.1.

C₁ C₂ C₃ 0 x₃x₄ x₆x₁ x₂x₅ 1 x₄x₅ x₀x₂ x₃x₆ 2 x₅x₆ x₁x₃ x₄x₀ 3 x₆x₀ x₂x₄ x₅x₁ 4 x₀x₁ x₃x₅ x₆x₂ 5 x₁x₂ x₄x₆ x₀x₃ 6 x₂x₃ x₅x₀ x₁x₄

Table 3.1: Color assignment of K₇

x0 x1 x2 x3 x4 x5 x6 4 5 6 0 1 2 3 4 5 6 0 1 2 3 5 6 0 1 2 3 4

C

C

C

3.2

Main Results

We begin this section with some notations. Let K_m(n) be the complete m-partite graph in which each partite set is of size n. In what follows, we will letZ_k ={1, 2, . . . , k} with the usual addition modulo k. For convenience, let V (K_m(n)) =

m−1 i=0

V_i where V_i =

{xi,0, xi,1,· · · , xi,n−1}. The graph Cm(n) is a spanning subgraph of V (Km(n)) where xi,j

is adjacent to x_i+1,k for all j, k ∈ Z_n and i ∈ Z_m (mod m). Clearly, if K_m can be decomposed into m−1₂ Hamiltonian cycles (m is odd), then K_m(n) can be decomposed into

m−1

2 subgraphs, each of which is isomorphic to Cm(n).

In order to prove the main theorem, we need the following two lemmas.

Lemma 3.2.1. Let p be an odd prime and m be a positive odd integer with p≤ m. Let

t∈ {1, 2, . . . , p − 1}. Then, there exists a set {S_i = (a_i,0, a_i,1, . . . , a_i,m−1)| 0 ≤ i ≤ p − 1}

(1) S₀ = (0, 0, . . . , 0, t);

(2) {a_i,j | 0 ≤ i ≤ p − 1} = {0, 1, 2, . . . , p − 1} j with 0 ≤ j ≤ m − 1; and (3) p w_i where w_i =

m−1 j=0

a_i,j for each i with 0≤ i ≤ p − 1.

Proof. The proof follows by direct constructions depending on the choice of t where 1 ≤ t ≤ p − 1. First, we let S₀ = (0, 0, . . . , 0, 1), S₁ = (1, 1, . . . , 1, 2), · · · , and S_p−1 = (p−1, p−1, . . . , p−1, 0) be the p m-tuples. For each i with 0 ≤ i ≤ p−1, let w_i =

m−1 j=0

a_i,j

where S_i = (a_i,0, a_i,1, . . . , a_i,m−1). If for each 0 ≤ i ≤ p − 1, p  w_i, we do nothing. Otherwise, assume that p | w_j for some j ∈ {1, 2, . . . , p − 1}, and note that such j is unique. Now, if j∈ {1, 2, . . . , p − 2}, replace S_j and S_j+1 with (j, j, . . . , j, j + 1, j + 1) and (j + 1, j + 1, . . . , j + 1, j, j + 2) respectively. Else, if j = p− 1, then replace S_p−2 and S_p−1 with (p− 2, p − 2, . . . , p − 2, p − 1, p − 1, p − 1) and (p − 1, p − 1, . . . , p − 1, p − 2, p − 2, 0) respectively.

When t = 1, clearly, these p m-tuples above satisfies all the four properties. So, in what follows, we consider 2 ≤ t ≤ p − 1. Note that we initially use the same m-tuples constructed in the case t = 1 and consider that j causing us to adjust entries above.

Case 1. No such j exists.

First, interchange a_0,m−1 with a_t−1,m−1. If w_t−1≡/ 0 (mod p), then we are done. On

the other hand, suppose w_t−1≡ 0 (mod p). If w_t≡/ 1 (mod p), then replace S_t−1and

S_t with (t−1, t−1, . . . , t−1, t, 1) and (t, t, . . . , t, t−1, t+1) respectively. Otherwise, replace S_t−1 and S_t with (t − 1, t − 1, . . . , t − 1, t − 1, t + 1) and (t, t, . . . , t, t, 1) respectively.

Case 2. j ∈ {1, 2, . . . , p − 2}.

First, interchange a_0,m−1 with a_t−1,m−1. If w_t−1≡/ 0 (mod p), then we are done. On

with (j, j, . . . , j, j + 1, j + 1, j + 1) and (j + 1, j + 1, . . . , j + 1, j, j, 1) respectively. Otherwise, interchange a_t−1,m−2 with a_t,m−2.

Case 3. j = p− 1.

Interchange a_0,m−1 with a_t−1,m−1.

Thus we can construct the desired p m-tuples.

Example 3.2.2. Take p = 5, m = 7. This implies that j = 2. Table 3.2 shows the structure of {S₀, S₁, S₂, S₃, S₄} for t = 1, 2, 3, and 4.

t = 1 t = 2 t = 3 t = 4 S₀ (0, 0, 0, 0, 0, 0, 1) (0, 0, 0, 0, 0, 0, 2) (0, 0, 0, 0, 0, 0, 3) (0, 0, 0, 0, 0, 0, 4) S₁ (1, 1, 1, 1, 1, 1, 2) (1, 1, 1, 1, 1, 1, 1) (1, 1, 1, 1, 1, 1, 2) (1, 1, 1, 1, 1, 1, 2) S₂ (2, 2, 2, 2, 2, 3, 3) (2, 2, 2, 2, 2, 3, 3) (2, 2, 2, 2, 2, 2, 1) (2, 2, 2, 2, 2, 3, 3) S₃ (3, 3, 3, 3, 3, 2, 4) (3, 3, 3, 3, 3, 2, 4) (3, 3, 3, 3, 3, 3, 4) (3, 3, 3, 3, 3, 2, 1) S₄ (4, 4, 4, 4, 4, 4, 0) (4, 4, 4, 4, 4, 4, 0) (4, 4, 4, 4, 4, 4, 0) (4, 4, 4, 4, 4, 4, 0) Table 3.2: Circulating sequences for p = 5 and m = 7

Lemma 3.2.3. Let v be a composite odd integer and p be the smallest prime with p|v.

Assume v = mp. If K_m admits an MHCP, then K_m(p) has a proper mp-edge-coloring that admits an MHCP.

Proof. We prove the lemma by giving a proper mp-edge-coloring ϕ. Since K_m defined on{x_i | i ∈ Z_m} admits an MHCP , let μ be such a proper edge-coloring using the colors 0, 1,· · · , m − 1. Let V (K_m(p)) =

m−1 i=0

V_i where V_i = {x_i,j | j ∈ Z_p} and L = [_h,k] be a circulant latin square of order p as defined before Figure 3.1. Now, we have a proper

mp-edge-coloring of K_m(p) by letting ϕ(x_a,bx_c,d) = μ(x_ax_c)· p + _b,d, where a, c ∈ Z_m and

b, d ∈ Z_p. Therefore, the edges in K_m(p) joining a vertex of V_a to a vertex of V_c, denoted (V_a, V_c), are colored with the entries in μ(x_ax_c)· p + L. It is not difficult to see that ϕ is a proper edge-coloring of K_m(p). Now, it is left to show that the edges of K_m(p) can be partitioned into multicolored Hamiltonian cycles.

Let C = (x_i0, x_i1,· · · , x_im−1) be a multicolored Hamiltonian cycle in K_m obtained from the M HCP of K_m. Define C_m(p) to be the subgraph induced by the set of edges in (V_i0, V_i1), (V_i1, V_i2), . . . , (V_im−2, V_im−1), (V_im−1, V_i0). Then, let S(r₀, r₁,· · · , r_m−1), where

r_j ∈ {0, 1, . . . , p − 1} for 0 ≤ j ≤ m − 1, be the set of perfect matchings in (V_i0, V_i1), (V_i1, V_i2), . . ., (V_im−2, V_im−1) and (V_im−1, V_i0), respectively, where the perfect matching in (V_ij, V_ij+1) is the set of edges x_ij,ax_ij+1,b with b− a ≡ r_j (mod p) for each j ∈ Z_m. Since these perfect matchings of (V_ij, V_ij+1) are multicolored, we have that S(r₀, r₁, . . . , r_m−1) is a multicolored 2-factor of C_m(n). Hence, we can partition the edges of C_m(p) into p multi-colored 2-factors due to the fact that r_i ∈ {0, 1, . . . , p − 1}. Note that S(r₀, r₁,· · · , r_m−1) and S(r₀, r₁,· · · , r_m−1 ) are edge-disjoint 2-factors if and only if r_i = r_i for each i∈ Z_m.

The proof follows by selecting (r₀, r₁,· · · , r_m−1) ∈ Zm_p properly in order that each 2-factor S(r₀, r₁,· · · , r_m−1) of C_m(p) is a Hamiltonian cycle. Observe that if

m−1



i=0

r_i is not a multiple of p (odd prime), then S(r₀, r₁,· · · , r_m−1) is a Hamiltonian cycle. From Lemma 3.2.1, let SS₀, SS₁,· · · , SS_p−1 be the 2-factors of C_m(p). This implies that we have a partition of the edges of C_m(p) into p edge-disjoint multicolored Hamiltonian cycles. Moreover, since K_m(p) can be partitioned into m−1₂ copies of C_m(p) where each C_m(p) arises from a multicolored Hamiltonian cycle in K_m, we have a partition of the edges of K_m(p) into m−1₂ · p multicolored Hamiltonian cycles.

Example 3.2.4. If m = p = 3, then the three multicolored Hamiltonian cycles are

S(0, 0, 1) = (x_0,0, x_1,0, x_2,0, x_0,1, x_1,1, x_2,1, x_0,2, x_1,2, x_2,2), S(1, 1, 2) = (x_0,0, x_1,1, x_2,2, x_0,1, x_1,2, x_2,0, x_0,2, x_1,0, x_2,1), S(2, 2, 0) = (x_0,0, x_1,2, x_2,1, x_0,2, x_1,1, x_2,0, x_0,1, x_1,0, x_2,2). In case that m = 5 and p = 3, then we have 6 multicolored Hamiltonian cycles. For each C₅₍₃₎, we have three multicolored Hamiltonian cycles of type S(0, 0, 0, 0, 1), S(1, 1, 1, 2, 2), and

S(2, 2, 2, 1, 0).

Following above example, in order to partition the edges of a 9-edge-colored K₉ into 4 Hamiltonian cycles, we combine S(0, 0, 1) with the three cliques (K₃) induced by the

three partite sets V₀, V₁ and V₂, to obtain a 4-factor. Since these K₃’s can be edge-colored with {3, 4, 5}, {6, 7, 8} and {0, 1, 2} respectively, we have a proper edge-colored 4-factor with each color occurs exactly twice. Thus, if this 4-factor can be partitioned into two multicolored Hamiltonian cycles, then we conclude that K₉ admits an M HCP . Figure 3.3 shows how this can be done.

0 1 2 3 8 4 5 7 6 4 5 3 6 7 8 0 1 2

x

x

x

x

x

x

x

x

x

Figure 3.3: Two multicolored Hamiltonian cycles in 9-edge-colored K₉

Notice that in the induced subgraphs < V₀ >, < V₁ > and < V₂ > we have exactly

one edge from each graph which is not included in the cycle with solid edges. Therefore, we may first color the edges in < V₀ >, < V₁ > and < V₂ > respectively and then adjust

the colors in (V₀, V₁), (V₁, V₂) and (V₂, V₀) respectively in order to obtain a multicolored Hamiltonian cycle. For example, if the color of x_0,0x_0,2 is 4 instead of 3, then we permute (or interchange) the two entries in 35 54 43

4 3 5 , and thus the latin square used to color

(V₁, V₂) becomes 45 53 34

3 4 5 . This is an essential trick we shall use when p is a larger

prime.

Before the following theorem, we introduce one useful notation. Let μ be a k-edge-coloring of a graph G. If K is a subgraph of G, for convenience, we use μ|_K to denote the edge-coloring of K induced by μ, i.e., μ|_K(e) = μ(e) for each e∈ E(K).