國
立
交
通
大
學
應用數學系
碩
士
論
文
圖的有向路徑覆蓋
Covering Graphs with Directed Paths
研 究 生：謝奇璁
指導教授：傅恆霖 教授
圖的有向路徑覆蓋
Covering Graphs with Directed Paths
研 究 生：謝奇璁 Student：Chi-Tsung Hsieh
指導教授：傅恆霖 Advisor：Hung-Lin Fu
國 立 交 通 大 學
應 用 數 學 系
碩 士 論 文
A Thesis
Submitted to Department of Applied Mathematics
College of Science
National Chiao Tung University
in Partial Fulfillment of the Requirements
for the Degree of
Master
in
Applied Mathematics
June 2009
Hsinchu, Taiwan, Republic of China
圖的有向路徑覆蓋
研究生：謝奇璁 指導老師：傅恆霖 教授
國 立 交 通 大 學
應 用 數 學 系
摘 要
在這篇論文裡，我們研究完全路徑雙覆蓋的有向形式。一個圖的有向路徑雙覆 蓋是在圖的對稱賦向裡的一個有向路徑集合，其中這個圖的對稱賦向裡的每一個 邊都要恰好出現在一個路徑裡，而且對圖裡的每一個點而言都會有唯一一條路徑 以此點當作起點以及會有唯一一條路徑以此點當作終點。在這篇論文中，首先我 們證明了如果一個圖形沒有包含連通部份為點數 3 的完全圖且為 3 退化圖則這個 圖就存在有向路徑雙覆蓋。再來我們也找出了完全二分圖 Kn,n與完全多分圖 Km(n)(n 為奇數,m≠3,5)的有向路徑雙覆蓋。Covering Graphs with Directed Paths
Student: Chi-Tsung Hsieh
Department of Applied Mathematics National Chiao Tung University
Hsinchu, Taiwan 30050
Advisor: Hung-Lin Fu
Department of Applied Mathematics National Chiao Tung University
Hsinchu, Taiwan 30050
Abstract
In this thesis we study an oriented version of perfect path double cover (PPDC). An oriented perfect path double cover (OPPDC) of a graph G is a collection of oriented paths in the symmetric orientation S(G) of G such that each edge of S(G) lies in exactly one of the paths and for each vertex v ∈ V (G) there is a unique path which begins in v (and thus the same holds also for terminal vertices of the
paths). First we show that if G has no components which isomorphism to K3
and G is a 3-degenerate graph, then G has an OPPDC. Next we also construct an
OPPDC for complete bipartite graph Kn,n and multipartite graph Km(n)(n is odd
and m 6= 3, 5),respectively.
謝誌
首先，我想感謝我的指導老師傅恆霖教授。在撰寫論文這段期間，我碰
到了許多瓶頸，傅老師總是能提供我一些精闢的意見，使我少走了許多冤
枉路。更重要的是，傅老師讓我在碩士生活裡除了學到做研究的方法與態
度之外，還學到許多為人處世的態度。
接著，我想要感謝陳秋媛老師、黃大原老師、以及翁志文老師。在我唸
研究所時對我的教導與鼓勵，除了解決我在課業上的煩惱與問題，也讓我
在唸研究所期間學到更多關於組合數學的知識。
特別感謝敏筠學姐與智懷學長，學長姐在我遇到問題的時候，總是不厭
其煩的指導我，讓我能更順利的完成論文研究。同時也感謝羅元勳、陳宏
賓、詹棨丰、張惠蘭、張雁婷等學長姐在研究方面對我的幫忙以及論文與
口試的建議。感謝研究所的各位同學，政軒、裴、軒軒、舜婷、偉帆、佩
純、子鴻、鈺傑等。感謝你們，讓我的研究所生活過的這麼愉快，謝謝。
最後，我要感謝我的家人，因為有你們一路上的支持與鼓勵，我才能在
碩士生活期間專心於課業上，也謝謝你們體諒我忙碌的生活，無法常回家
陪伴你們。在此謝謝爸爸、媽媽、哥哥、弟弟，謝謝你們的支持讓我順利
完成學業，謝謝。
Contents
Abstract (in Chinese) i
Abstract (in English) ii
Acknowledgement iii Contents iv 1 Introduction 1 1.1 Preliminaries . . . 3 1.2 Known Results . . . 6 2 Main Results 12 2.1 3-degenerate graph . . . 12 2.2 OPPDC on Kn,n and Km(n) . . . 15 3 Conclusion 18 References 19
1
Introduction
Graph decomposition is one of the most important topics in the study of graph theory.In 1979, P. D. Seymour [20] conjectured that every bridgeless graph has a cycle double cover, which is a collection of cycles such that every edge of G is contained in exactly two cycles of the cycle collection. The cycle double cover conjecture lies in the very heart of the graph theory. It seems that this elementary problem has a deep topological background and only partial results are known. This problem (in the very short time of its existence) also motivated several related conjectures: J. A. Bondy [3] conjectured that every simple bridgeless graph has a small cycle double cover, which is a cycle double cover containing at most n − 1 cycles on a graph that order n. There are a number of classes of graphs for which the small cycle double cover conjecture has been verified, including complete
graphs [3] (excluding K2), complete bipartite graph [3] (other than K1,m), 4-connected
planar graphs [18], and simple triangulations of orientable surfaces [3, 17]. A common characteristic of these classes of graphs is that there is some structure to the graphs that allows for assumptions about cycles in the graphs. This seems to be a desirable property, since it is necessary to keep track of the number of cycles when constructing small cycle double covers.
In 1990, Bondy [3] also posed several conjectures about path double covers of graphs. He conjectured that every simple graph admits a path double cover P such that each vertex occurs exactly twice as an end of a path in P : a perfect path double cover. This conjecture was later provey by H. Li [11]. Bondy also conjectured that every k-regular simple graph admits a path double cover P such that every path in P has length k and each vertex of the graph occurs exactly twice as an end of a path in P : a regular perfect path double cover. This conjecture has been proved for k ≤ 3 [3] and k = 4 [8] but is still open for larger values of k. Perfect path double cover for graphs in general is equivalent to small cycle double cover for bridgeless apex graphs (apex graph = graph with a vertex joined to all other vertices). To see this, consider a graph G\v where v is a vertex of degree n − 1 in a bridgeless graph G. G has an small cycle double cover if and only if G\v has a perfect path double cover.
Also unsolved are oriented versions of these problems. In 1988, Jaeger [9] conjectured that every bridgeless graph has an oriented cycle double cover. No counterexample to the oriented cycle double cover conjecture is presently known. In 1998, J. Maxov´a [12] show
that K3 and K5 have no oriented perfect path double cover. In 2001, J. Maxov´a proved
that all 2-connected graph on n vertices with at most 2n−1 edges have an oriented perfect
path double cover(except for K3). In 2004, J. Maxov´a conjectured that K3 and K5 are
the only connected graphs which do not have an oriented perfect path double cover. In this thesis, the main results are that for every 3-degenerate graph with no
compo-nents isomorphic to K3 has an oriented perfect path double. Furthermore, show that for
all n ≥ 1 the complete bipartite graph Kn,n has an oriented perfect path double and for
m 6= 3, 5 and n is odd the multipartite graph Km(n) has an oriented perfect path double.
1.1
Preliminaries
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.[23]
A graph G is a triple consisting of a vertex set V (G), an edge set E(G), and a relation that associates with each edge 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 or 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.
If e = {u, v} (uv in short) is an edge of G, then e is said to be incident to u and v. 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, deg(v) = |N(v)|, is the number of neighbors of v.
Let G = (V ; E) be a undirected simple graph. A path of length k in G is a sequence
v1, e1, v2, . . . , ek, vk+1 of its vertices and edges where ei = {vi, vi+1} for 0 ≤ i ≤ k and
v1, . . . , vk+1 are distinct vertices. A cycle of length k is a sequence v1, e1, v2, . . . , ek, vk+1
of its vertices and edges where ei = {vi, vi+1} for 0 ≤ i ≤ k, v1 = vk+1 and v1, . . . , vk are
distinct vertices.
The maximum degree is △(G), the minimum degree is δ(G), and G is regular if △(G) = δ(G). It is k-regular if the common degree is k. A cubic graph is a graph that is regular of degree 3.
A graph G is connected if it has a u, v-path whenever u, v ∈ V (G) (otherwise, G is disconnected). If G has a u, v-path, then u is connected to v in G. The components of a graph G are its maximal connected subgraphs. A component (or graph) is trivial if it has no edges; otherwise it is nontrivial. An isolated vertex is a vertex of degree 0.
A directed graph (digraph) is a triple consisting of a vertex set V (G), an edge set E(G), and a function assigning each edge an ordered pair of vertices. The first vertex of the ordered pair is the tail of the edge, and the second is the head; together they are the endpoints. We say that an edge from its tail to its head.
A digraph is a path if it is a simple digraph whose vertices can be linearly ordered so that there is an edge with tail u and head v if and only if v immediately follows u in the vertex ordering. A cycle is defined similarly using an ordering of the vertices on a circle.
In our main results, all graph we consider are simple digraph.
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. A spanning subgraph of G is a subgraph H with V (H) = V (G). A graph G is k − degenerate if every subgraph of G has a vertex of degree at most k.
A complete graph is a simple graph whose vertices are pairwise adjacent; the complete
graph with n vertices is denoted by Kn. 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 bipartite 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 Ks,t. If
the sets have the same size n, the complete bipartite graph is called balanced, which is
denoted by Kn,n. Similarly, the complete m-partite graph is denoted by Ks1,s2,...,sm and
the balanced complete m-partite graph is denoted by Km(n) where each partite set has n
vertices.
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.
Let G be a graph of order m with V (G) = {gi : 0 ≤ i ≤ m − 1}, and let H ne a
graph of order n with V (H) = {hi : 0 ≤ i ≤ n − 1}. The Cartesian product GH is
defined to be the graph with vertex set {(gi, hj) : 0 ≤ i ≤ m − 1 and 0 ≤ j ≤ n − 1} and
(gi, hj)(gs, ht) ∈ E(GH) if either gi = gs and hjht ∈ E(H) or hj = ht and gigs ∈ E(G).
The symmetric orientation of G, denoted by S(G), that is an oriented graph obtained from G by replacing each edge of G by a pair of oppositely directed arcs (i.e. V (S(G)) = V (G) and E(S(G)) = {(u, v), (v, u)|(u, v) ∈ E(G)}).
We give some important definitions as followings.
A cycle double cover (CDC) of a graph G is a collection of its cycle such that each edge of G lies in exactly two of the cycles. A small cycle double cover (SCDC) of a graph on n vertices is a CDC with at most n − 1 circuits.
A perfect path double cover (PPDC) of a graph G is a collection of its paths such that each edge of G lies in exactly two of the paths and each vertex of G appears precisely twice as an endpoint of a path.
A regular perfect path double cover (RPPDC) of a k-regular simple graph G is a col-lection P of its paths such that every path in P has length k and each vertex of the graph occurs exactly twice as an end of a path in P.
For a path double cover P of a graph G, the associated graph AP(G) of P is defined as
a graph having the same vertex set as G, with two vertices x and y adjacent in AP(G) if
and only if there is a path in P with endpoints x and y.
A PPDC is called an eulerian perfect path double cover (EPPDC) if its associated graph is a cycle. If a path double cover is both eulerian and regular, we call it an ERPPDC.
An oriented perfect path double cover (OPPDC) of a graph G is a collection of paths on G such that each edge of S(G) lies in exactly one of the paths and each vertex of G appears just once as a beginning and just once as an end of a path.
1.2
Known Results
We consider cycle decomposition and path decomposition on undirected graph. The following are some results:
Theorem 1.1. [10] (1) For all odd integers n and all non-negative integer r satisfying 3r = n(n−1)2 there is a decomposition of Kn into r 3-cycles which partitions the edge set of
Kn. (2) For all even integers n and all non-negative integers r satisfying 3r = n(n−2)2 there
is a decomposition of Kn− F into r 3-cycles which partitions the edges set of Kn− F .
We can establish the existence of cycle systems not only the 3-cycle system but also the m-cycle system for any m. There are some results below:
Theorem 1.2. [16] (1) For all odd integers n and all non-negative integer r and m
satisfying mr = n(n−1)2 there is a decomposition of Kn into r m-cycles which partitions the
edge set of Kn. (2) For all even integers n and all non-negative integers r and m satisfying
mr = n(n−2)2 there is a decomposition of Kn− F into r m-cycles which partitions the edges
set of Kn− F .
Theorem 1.3. [1] (1) For all odd integers n and all non-negative integer r and s satisfying
3r + 5s = n(n−1)2 there is a decomposition of Kn into r 3-cycles and s 5-cycles which
partitions the edge set of Kn. (2) For all even integers n and all non-negative integers r
and s satisfying 3r + 5s = n(n−2)2 there is a decomposition of Kn− F into r 3-cycles and
s 5-cycles which partitions the edges set of Kn− F .
Theorem 1.4. [7] (1) For all odd integers n and all non-negative integer r, s and t
satisfying 3r + 4s + 6t = n(n−1)2 there is a decomposition of Kn into r 3-cycles, s 4-cycles,
and t 6-cycles which partitions the edge set of Kn. (2) For all even integers n and all
non-negative integers r, s and t satisfying 3r + 4s + 6t = n(n−2)2 there is a decomposition of
Kn− F into r 3-cycles, s 4-cycles, and t 6-cycles which partitions the edges set of Kn− F .
Theorem 1.5. [4] (1) For all odd integers n and all non-negative integer r and s satisfying
4r + 5s = n(n−1)2 there is a decomposition of Kn into r 4-cycles and s 5-cycles which
partitions the edge set of Kn. (2) For all even integers n and all non-negative integers r
and s satisfying 4r + 5s = n(n−2)2 there is a decomposition of Kn− F into r 4-cycles and
s 5-cycles which partitions the edges set of Kn− F .
The following useful contains three different lengths which are n, n − 1, n − 2.
Theorem 1.6. [7] Let S = {n−2, n−1, n}. If n is odd and a(n−2)+b(n−1)+cn = n(n−1)2 ,
then Kn = aCn−2+ bCn−1+ cCn. If n is even and a(n − 2) + b(n − 1) + cn = n(n−2)2 , then
Kn− F = aCn−2+ bCn−1+ cCn.
Alspach Conjecture is also true if the cycles lengths mi are bounded by some linear
function of n and n is sufficiently large.
Theorem 1.7. [2] Assume n must be larger than N2 which is very large absolute constants.
If m1, . . . , mt are integers with 3 ≤ mi ≤ ⌊n−112120 ⌋ andPi=1t mi = (n2) (n odd) or (n2) −n2 (n
even), then one can pick Kn (n odd) or Kn− I(n even) with cycles of length m1, . . . , mt.
Theorem 1.8. [6] Let n be a n even positive integer. Then Kn can be decomposed into
n
2hamiltonian paths.
Theorem 1.9. [15] If n is odd and {ai : 1 ≤ i ≤ r} is a multiset of r positive integers
satisfying 1 ≤ ai ≤ n − 2 and Pri=1ai = (n2). Then Kn can be decomposed into {Pai|1 ≤
i ≤ r}.
Theorem 1.10. [21] Let m|λ(n
2), and m ≤ n − 1. Then λKn caon be decomposed into
isomorphic paths of length m.
Theorem 1.11. [5] If v is odd. Let m1, m2, . . . , mt be t positive integers such that 1 ≤
mi ≤ n − 2, Pti=1mi + k(n − 1) = (n2), and k ∈ {1, 2,n−12 }, then Kv can be decomposed
into t + k paths P1, P2, . . . , Pt+k such that the length of Pi is mi for i = 1, 2, . . . , t and
the length of Pi is n − 1 for i > t.
Theorem 1.12. [5] If v is odd. Let n − 1 ≥ m1 ≥ m2 ≥ · · · ≥ mt ≥ 1 and h ≤
mt ≤ n − h − 1 such that Pti−1mi = (n2), m1 = m2 = · · · = mh = n − 1. Then
Kv can be decomposed into t paths P1, P2, . . . , Pt such that the length of Pi is mi for
i = 1, 2, . . . , t. Moreover, if there exists a h < t′ ≤ t such that h ≤ m
h ≤Pt
i=t′mi ≤ n − h − 1, then Kv can be decomposed into t paths P
1, P2, . . . , Pt such
that the length of Pi is m
i for i = 1, 2, . . . , t.
Theorem 1.13. [5] If v is odd. Let n − 1 ≥ m1 ≥ m2 ≥ · · · ≥ mt ≥ 1, mt < h, and
mt−1mt ≤ n − h − 1 such that Pi=1tmi = (n2), m1 = m2 = . . . = mh = n − 1. Then
Kv can be decomposed into t paths P1, P2, . . . , Pt such that the length of Pi is mi for
i = 1, 2, . . . , t.
Theorem 1.14. [5] If v is odd. Let n−1 ≥ m1 ≥ m2 ≥ · · · ≥ mt≥ 1 and n+h−2 ≤ mt+
mt−1 ≤ 2n−h−3 such thatPti−1mi = (2n), m1 = m2 = . . . = mh = n−1. Then Kv can be
decomposed into t paths P1, P2, . . . , Pt such that the length of Pi is m
i for i = 1, 2, . . . , t.
Moreover, if there exists a h < t′ ≤ t such that n + h − 2 ≤ Pt
i=t′mi ≤ 2n − h − 3,
then Kv can be decomposed into t paths P1, P2, . . . , Pt such that the length of Pi is mi
for i = 1, 2, . . . , t.
Next, we consider some results of RPPDC and EPPDC.
Proposition 1.15. [19] Suppose that G is a graph with an eulerian perfect path double cover. Then for 1 ≤ d(x) ≤ 3, G + x has an eulerian perfect double cover.
Proposition 1.16. [19] For any n ≥ 1, Kn,n has an RPPDC. Moreover, if n is odd, then
Kn,n has an ERPPDC.
Proposition 1.17. [19] For any m, n ≥ 1, Km,n has an EPPDC.
Proposition 1.18. [19] If G is a k-regular graph, k ≥ 1, then L(G) has an RPPDC. Proposition 1.19. [19] Let G be a graph with m edges. Suppose 2G has an Euler circuit
e1, e2, . . . , e2m such that S1 = {e1, e3, . . . , e2m−1} and S2 = {e2, e4, . . . , e2m} are both the
set E(G) of all edges of G. Furthermore, suppose that for each v ∈ V (G) there is ordering, C(v), of the edges incident to v such that every pair of consecutive edges inC(v) occurs exactly once as a pair of consecutive edges in the Euler circuit. Then L(G) has an EPPDC.
Proposition 1.20. [19] For all m ≥ 2, L(Km) has an ERPPDC.
Proposition 1.21. [19] For all m, n ≥ 1, L(Km,n) has an RPPDC. Furthermore, if
gcd(n, m) = 1 or gcd(n, n − m + 2) = 1, then L(Km,n) has an ERPPDC.
Proposition 1.22. [19] For every positive odd integer n, L(Kn,n) has an ERPPDC.
Proposition 1.23. [19]
• If G and H have RPPDCs, then GH has an RPPDC.
• If G and H have EPPDCs and (|G|, |H|) = 1, then GH has an EPPDC.
Proposition 1.24. [19] If G has an EPPDC, then the Cartesian product GK2 has an
EPPDC.
Proposition 1.25. [19] For all n ≥ 0, the n-cube, Qn, has an EPPDC.
In our main result we concentrate on oriented version. Now, we consider some results below:
Lemma 1.26. [13] K2n has an OPPDC.
Proof. Let V (K2n) = {v0, v1, . . . , v2n−1}. For 0 ≤ i ≤ 2n−1 set Pi = (vi, vi+1, vi−1, vi+2, vi−2, . . . , vi+n)
where all subscripts are read modulo 2n. It is easy to verity that P = {Pi|0 ≤ i ≤ 2n − 1}
is an OPPDC of K2n.
Lemma 1.26 gives an easy construction of OPPDC for all K2n. Tillson proved in [22]
that all K2n+1 have an OPPDC for n ≥ 3.
Example 1.27. [13] K7 has an OPPDC as follow:
P1 = 1263547 P2 = 2731465
P3 = 3742516 P4 = 4536721
P5 = 5764132 P6 = 6175243
P7 = 7156234
We can check that the collection P = {P1, . . . , P7} is an OPPDC of K7.
Next, we consider the minimal (i.e. , with minimal number of edges) connected graph
G such that G 6= K3, G 6= K5 and G has no OPPDC. J. Maxov´a had show that G has
Lemma 1.28. [13] Let G1, G2 be two graphs which have an OPPDC. Suppose that G1∩
G2 = {v}.Then the union G1∪ G2has an OPPDC.
Proof. Denote by Pian OPPDC of Gi, i = 1, 2. Let P1 ∈ P1be the path that starts in v
and P2 ∈ P2be the path that ends at v. Then the collection P1∪ P2∪ {P1∪ P2}\{P1, P2}
is an OPPDC of G1∪ G2.
Corollary 1.29. [13] Let G be a simple graph; G 6= K3, and v ∈ V {G} a vertex of degree
1. If G \ v has an OPPDC then G has an OPPDC.
By applying this corollary, we get that if we add a new vertex of degree 1 to a graph with an OPPDC then the resulting graph also has an OPPDC. Hence every tree has an OPPDC.
Theorem 1.30. [13] Let G be a simple graph; G 6= K3,and v ∈ V {G} a vertex of degree
2. If G \ v has an OPPDC then G has an OPPDC.
By applying Corollary 1.29 and Theorem 1.30, we get that every 2-degenerate graph
has an OPPDC, except K3. The following are some results
Corollary 1.31. [13] If G is a union of two arbitrary trees; G 6= K3; then G has an
OPPDC.
Another construct which preserves the property of having an OPPDC is the so-called arrow construction.
Definition 1.32. [14] A graph I with two distinguished vertices a, b, a, b /∈ E(I), is called
an indicator. For a given directed graph D = (V, E) and an indicator (I, a, b) we define an (undirected) graph D ∗ (I, a, b) = (W, F ) sa follows:
W = (E × V (I))/ ∼,
where the equivalence is generated by the following pairs:
((x, y), a), ((x, y′), a), ((x, y), b), ((x′, y), b), ((x, y), b), ((y, z), a).
For a pair (e, x) ∈ E × V (I) its equivalence class is denoted by [e, x].
We put {[e, x], [e′, x′]} ∈ F ⇐⇒ e = e′ and {x, x′} ∈ E(I).
Figure 1: Arrow construction
This arrow construction is schematically indicated in Fig. 1 (One can check that the indicator I in Fig. 1 satisfies the assumptions of Theorem1.33 below.)
Theorem 1.33. [14] Suppose an indicator (I, a, b) has an OPPDC Π containing two
paths P1, P2 ∈ Π such that P1 begins in a and ends in b, and P2 begins in b and ends
in a. Further suppose G has an OPPDC. Then for any orientation D of G the graph D ∗ (I, a, b) has an OPPDC.
Proposition 1.34. [14] If G is a 2-connected graph with |E(G)| ≤ 2n − 1; G 6= K3; then
G has an OPPDC.
Conjecture 1.35. [14] K3 and K5 are the only connected graphs which do not have an
2
Main Results
In this section, we focus on the minimal degree of a graph G first. We show that if we add a new vertex of degree 3 to a graph with an OPPDC then the resulting graph also has an OPPDC. And we use this theorem to prove that if G is a 3-degenerate graph and G has
no components which isomorphism to K3 then G has an OPPDC. Next, we show that
the complete graph Kn,n and the multipartite graph Km(n) has an OPPDC by a special
construction.
2.1
3-degenerate graph
Theorem 2.1. Let G be a simple graph; G 6= K3,and v ∈ V {G} a vertex of degree 3. If
G \ v has an OPPDC then G has an OPPDC.
Proof. Let N(v) = {a, b, c} be the neighbors of the vertex v. Denote by P an OPPDC
of the graph G \ v. For u ∈ V (G \ v), letPu (resp. P
u) denote the path of P beginning
(resp. ending) with u. We call Pu (resp. P
u) is the outer (resp. inner) path of u in G \ v.
Case 1. There exists an outer path Pu,u ∈ N(v),Pu pass through N(v).
Without loss of generality , we assume Pa pass through b and then c.
Subcase 1-1: Pc 6= Pb
Separate Pa into two paths, P
1 and P2,where P1 is the path that beginning at a
and ending at b along Pa and P
2 is the path that beginning at b and along Pa.
Let Pc∗= (c, v) ∪ (v, a) ∪ P 1 Pa∗ = (a, v) ∪ (v, b) ∪ P 2 Pv∗ = Pb∪ (b, v) Pv∗ = (v, c) ∪ Pc
Then the collection P \ {Pa, P
b, Pc} ∪ {Pv∗, Pv∗, Pc∗, Pa∗}is an OPPDC of G.
Subcase 1-2: Pc = Pb
Subcase 1-2-1: Pb 6= P
a
Separate Pa into three paths , P
1, P2, and P3, where P1 is the path that
beginning at a and ending at b along Pa , P
2 is the path that beginning at b
and ending at c along Pa , P
3 is the path that beginning at c and along Pa.
Let Pa∗ = P 1 ∪ (b, v) ∪ (v, c) ∪ P3 Pb∗ = P 2∪ (c, v) ∪ (v, a) Pv∗= (v, b) ∪ Pb Pv∗ = Pa∪ (a, v)
Then the collection P \ {Pa, Pa, Pb, } ∪ {Pa∗, Pb∗, Pv∗, Pv∗} is an OPPDC of
G.
Subcase 1-2-2: Pb = P
a
Let Pa∗ = (a, v) ∪ (v, c) ∪ Pc
Pc∗ = (c, v) ∪ (v, b) ∪ Pb
Pv∗= (v, a) ∪ Pa
Pv∗ = (b, v)
Then the collection P \ {Pa, Pb, Pc} ∪ {Pa∗, Pc∗, Pv∗, Pv∗} is an OPPDC of G.
Case 2. There is no outer path Pu,u ∈ N(v),Pu pass through N(v).
Without loss of generality , we assume Pa doesn’t pass through c.
Subcase 2-1: Pc doesn’t pass through a.
Let Pc∗= (c, v) ∪ (v, a) ∪ Pa
Pa∗ = (a, v) ∪ (v, c) ∪ Pc
Pv∗ = (v, b) ∪ Pb
Pv∗ = (b, v)
Then the collection P \ {Pa, Pb, Pc} ∪ {Pa∗, Pc∗, Pv∗, Pv∗} is an OPPDC of G.
Subcase 2-2: Pc pass through a.
Since Pc passes through a, it can’t pass through b. If Pb doesn’t pass through c,
it will return to case 2-1. So Pb passes through c. If Pb passes through a, it will
return to case 1. So Pb doesn’t pass through a.
Let Pc∗= (c, v) ∪ (v, a) ∪ Pa
Pa∗ = (a, v) ∪ (v, b) ∪ Pb
Pv∗ = (v, c) ∪ Pc
Then the collection P \ {Pa, Pb, Pc} ∪ {Pa∗, Pc∗, Pv∗, P
v∗} is an OPPDC of G.
Thus, we have the prove.
Lemma 2.2. G1 and G2 has an OPPDC.
Proof. Since degG1(V1) = 3 degG1(V2) = 2 and G1\V1, G1\V1 are paths. By Theorem
1.30 and Theorem 2.1 we know that G1 and G2 has an OPPDC.
Theorem 2.3. If G has no components which isomorphism to K3 and G is a 3-degenerate
graph, then G has an OPPDC.
Proof. We proceed by induction on n = V |(G)|. Since G is a 3-degenerate graph,
there is a vertex v ∈ V (G) of degree at most 3. We denote that G′ = G \ v. If G′ is
isomorphic to K3 then G is isomorphic to K4 or one of the graphs G1, G2 in Lemma 2.2,
that all have an OPPDC. If G′ is a disconnected graph with some components which are
isomorphic to K3. Then we choose another vertex v′ which in K3 and let G′ = G \ v′.
Since degG(v′) ≤ 3 we know that G′ applies to the induction hypothesis. If degG(v) = 1
by induction hypothesis the graph G′ has an OPPDC. Then by applying Corollary 1.29
the graph G has an OPPDC. If degG(v) = 2 by induction hypothesis the graph G′ has an
OPPDC. Then by applying Theorem 1.30 the graph G has an OPPDC. If degG(v) = 3
by induction hypothesis the graph G′ has an OPPDC. Then by applying Theorem 2.1 the
graph G has an OPPDC. Thus, we have the prove.
Corollary 2.4. Every cubic graph has an OPPDC.
Proof. We know that every cubic graph is 3 − degenerate. By Theorem 2.3 we have the prove.
2.2
OPPDC on K
n,nand K
m(n)Now, we consider a special construction of OPPDC on complete graph Kn,n and
multi-partite graph Km(n).
Lemma 2.5. For all n ≥ 1, Kn,n has an OPPDC.
Proof. Assume that Kn,n has bipartition (X, Y ), where X = {x0, x1, . . . , xn−1} and
Y = {y0, y1, . . . , yn−1}. Suppose that n is odd. For each i, i = 0, 1, . . . , n − 1, let Pi =
(xi, yn−1+i, xi+1, yn−2+i, . . . , y(n−3)/2+i, x(n−1)/2+i, y(n−1)/2+i). Then P = {P0, P1, . . . , Pn−1}
is a path decomposition of Kn,n. Let Pi′ = (y(n−1)/2+i, x(n−1)/2+i, y(n−3)/2+i, . . . , xi+1, yn−1+i, xi),
and let P′ = {P
0′, P1′, . . . , Pn−1′} .The union of the two path decompositions forms an
OPPDC of Kn,n.
If n is even, let Pi = (xi, yn−1+i, xi+1, yn−2+i, . . . , yn/2+i, xn/2+i). Then P = {P0, P1, . . . , Pn−1}
is a path decomposition of Kn,n. Exchanging the x′s and y′s we obtain a second path
decomposition P′ of K
n,n. The union of these two path decompositions forms an OPPDC
of Kn,n.
Example 2.6. An OPPDC of K5,5.
Assume that K5,5 has bipartition (X, Y ), where X = {x0, x1, x2, x3, x4} and Y =
{y0, y1, y2, y3, y4}. P = (x0, y4, x1, y3, x2, y2) (x1, y0, x2, y4, x3, y3) (x2, y1, x3, y0, x4, y4) (x3, y2, x4, y1, x0, y0) (x4, y3, x0, y2, x1, y1) P′ = (y2, x2, y3, x1, y4, x0) (y3, x3, y4, x2, y0, x1) (y4, x4, y0, x3, y1, x2) (y0, x0, y1, x4, y2, x3) (y1, x1, y2, x0, y3, x4) Then P ∪ P′ is an OPPDC of K 5,5. Example 2.7. An OPPDC of K4,4.
Assume that K5,5has bipartition (X, Y ), where X = {x0, x1, x2, x3} and Y = {y0, y1, y2, y3}.
P = (x0, y3, x1, y2, x2) (x1, y0, x2, y3, x3) (x2, y1, x3, y0, x0) (x3, y2, x0, y1, x1) P′ = (y0, x3, y1, x2, y2) (y1, x0, y2, x3, y3) (y2, x1, y3, x0, y0) (y3, x2, y0, x1, y1)
Then P ∪ P′ is an OPPDC of K 4,4.
Theorem 2.8. Km(n) has an OPPDC for n is odd, m 6= 3, 5.
Proof. Suppose that m is even. Let V (Km(n)) =
2k−1
[
i=0
Vi where Vi = {xi,0, xi,1, · · · , xi,n−1}
and m = 2k. By Lemma 1.26 K2k has an OPPDC, and by Lemma 2.5 Kn,n has an
OPPDC. Then we let Q= 2k−1 [ i=0 (P0
i,i+1+ Pi+1,i−10 + Pi−1,i+20 + . . . + Pi−k+1,i+k0 )
∪ (P1
i,i+1+ Pi+1,i−1n−1 + Pi−1,i+21 + . . . + Pi−k+1,i+k1 )
∪ (P2
i,i+1+ Pi+1,i−1n−2 + Pi−1,i+22 + . . . + Pi−k+1,i+k2 )
... ∪ (Pi,i+1n−1 + P1
i+1,i−1+ Pi−1,i+2n−1 + . . . + Pi−k+1,i+kn−1 )
where Pi,jq = (yi,q, xj,n−1+q, yi,q+1, xj,n−2+q, . . . , xj,n−3
2 +q, yi, n−1
2 +q, xj, n−1
2 +q)
and yi,q= fj,i(xi,q) by
f : yi,n−1 2 −j = xi,j, if j < n−1 2 yi,0 = xi,j, if j = n−12 yi,3n−1 2 −j = xi,j, if j > n−1 2 . Then Q is an OPPDC of Km(n).
Now, we consider m is odd, m 6= 3, 5. Let P = {P0, P1, . . . , Pm−1} is an OPPDC of Km
and denote that Pi = (vi(0), vi(1), . . . , vi(n − 1)), where vi(0) is the beginning at the path
Pi and vi(n − 1) is the end at the path Pi.
Then we let R= m−1 [ i=0 (P0
i(0),i(1)+ Pi(1),i(2)0 + Pi(2),i(3)0 + . . . + Pi(n−2),i(n−1)0 )
∪ (P1
i(0),i(1)+ Pi(1),i(2)n−1 + Pi(2),i(3)1 + . . . + Pi(n−2),i(n−1)1 )
∪ (P2
i(0),i(1)+ Pi(1),i(2)n−2 + Pi(2),i(3)2 + . . . + Pi(n−2),i(n−1)2 )
... ∪ (Pi(0),i(1)n−1 + P1
i(1),i(2)+ Pi(2),i(3)n−1 + . . . + Pi(n−2),i(n−1)n−1 )
where Pi,jq = (yi,q, xj,n−1+q, yi,q+1, xj,n−2+q, . . . , xj,n−3
2 +q, yi, n−1
2 +q, xj, n−1
2 +q)
and yi,q= fj,i(xi,q) by
f = xi,j → yi,n−1 2 −j, if j < n−1 2 xi,j → yi,0, if j = n−12 xi,j → yi,3n−1 2 −j, if j > n−1 2 . Then R is an OPPDC of Km(n). 16
Example 2.9. An OPPDC of K4(3).
Let V (K4(3)) =
3
[
i=0
Vi where Vi = {xi,0, xi,1, xi,2}. Let
Q= 3 [ i=0 (P0
i,i+1+ Pi+1,i−10 + Pi−1,i+20 )
∪ (P1
i,i+1+ Pi+1,i−12 + Pi−1,i+21 )
∪ (P2
i,i+1+ Pi+1,i−11 + Pi−1,i+22 )
where Pi,jq = (yi,q, xj,2+q, yi,1+q, xj,1+q) and yi,q = fj,i(xi,q) by
f : yi,1 = xi,0, yi,0 = xi,1, yi,2 = xi,2. ⇒ Q= (x0,1, x1,2, x0,0, x1,1, x3,2, x1,0, x3,1, x2,2, x3,0, x2,1) (x0,0, x1,0, x0,2, x1,2, x3,1, x1,1, x3,0, x2,0, x3,2, x2,2) (x0,2, x1,1, x0,1, x1,0, x3,0, x1,2, x3,2, x2,1, x3,1, x2,0) [ (x1,1, x2,2, x1,0, x2,1, x0,2, x2,0, x0,1, x3,2, x0,0, x3,1) (x1,0, x2,0, x1,2, x2,2, x0,1, x2,1, x0,0, x3,0, x0,2, x3,2) (x1,2, x2,1, x1,1, x2,0, x0,0, x2,2, x0,2, x3,1, x0,1, x3,0) [ (x2,1, x3,2, x2,0, x3,1, x1,2, x3,0, x1,1, x0,2, x1,0, x0,1) (x2,0, x3,0, x2,2, x3,2, x1,1, x3,1, x1,0, x0,0, x1,2, x0,2) (x2,2, x3,1, x2,1, x3,0, x1,0, x3,2, x1,2, x0,1, x1,1, x0,0) [ (x3,1, x0,2, x3,0, x0,1, x2,2, x0,0, x2,1, x1,2, x2,0, x1,1) (x3,0, x0,0, x3,2, x0,2, x2,1, x0,1, x2,0, x1,0, x2,2, x1,2) (x3,2, x0,1, x3,1, x0,0, x2,0, x0,2, x2,2, x1,1, x2,1, x1,0) Then Q is an OPPDC of K4(3).
3
Conclusion
In this thesis, we have obtained the following main results:
1. If G has no components which isomorphism to K3 and G is a 3-degenerate graph,
then G has an OPPDC.
2. For all n ≥ 1, Kn,n has an OPPDC.
3. Km(n) has an OPPDC for n is odd, m 6= 3, 5.
But, we are still far from verifying the conjectures(Conjecture 1.35). Hopefully, this task can be done in the near future.
References
[1] P. Adams, D. E. Bryant and A. Khodkar, (3,5)-cycles decompositions. J. Combin. Designs, 6 (1998), 91-110.
[2] P. N. Balister. On the Alspach conjecture. Combin., Probability and computing, 10 (2001), 95-125.
[3] J. A. Bondy, Small cycle double cover of graphs, in: G. Hahn, G. Sabidussi, R.Woodrow (Eds.), Cycles and Rays, NTAO ASI Series C, Kluwer Academic Pub-lishers, Dordrecht, 1990, pp. 21-40.
[4] D. E. Bryant, A. Khodkar and H. L. Fu, (m,n)-cycles systems. J. Statist, Planning and Inference, 74 (1998), 365-370.
[5] P. K. Chuang, Decomposing Complete Graph into Paths with Prescribed Lengths, M. Sc. Thesis, National Chiao Tung University, 2003.
[6] F. Harary, Graph Theory, Addison-Wesley, Reading MA, 1972.
[7] K. Heinrich, P. Horak, A. Rosa, On Alspachs conjecture, Discrete Math., 77 (1989), 97-121.
[8] K. Heinrich, P. Horak, W.D. Wallis, Q. Yu, Perfect double covers with paths of length four, J. Graph Theory 21 (1996) 187-197.
[9] F. Jaeger, Nowhere-zero Pow problems, in: L.W. Beineke, R.J. Wilson (Eds.), Se-lected Topics in Graph Theory 3, Academic Press, London, 1988, pp. 71V95.
[10] Rev. T. P. Kirkman, On the problem in combinations, Cambr. and Dublin Math. J., 2 (1847), 191-204.
[11] H. Li, Perfect path double covers in every simple graph, J. Graph Theory 14 (1990) 645-650.
[12] J. Maxov´a, Oriented perfect path double cover, Diploma Thesis, Charles University,
[13] J. Maxov´a, J. Nesetril, On oriented path double covers, Discrete Math. 233 (2001) 371-380.
[14] J. Maxov´a, J. Nesetril, On covers of graphs, Discrete Math. 276 (2004) 287-294.
[15] K. W. P. Ng, On Path decompositions of Complete Graphs, M. Sc. Thesis, Simon Fraser University, 1985.
[16] M. Sajna, Cycle decompositions III : complete graphs and fixed length cycles, J. Combin. Designs, 10 (2002), 27-78.
[17] K. Seyffarth, Maximal planar graphs of diameter two, J. Graph Theory 13 (1989) 619V648.
[18] K. Seyffarth, Small cycle double covers of 4-connected planar graphs, Combinatorica 13 (1993) 477V482.
[19] K. Seyffarth, Chengde Wang, On eulerian and regular perfect path double covers of graphs, Discrete Math. 293 (2005) 237-250.
[20] P. D. Seymour, Sums of circuits, in: J.A. Bondy, U.S.R. Murty (Eds.), Graph Theory and Related Topics, Academic Press, New York, 1979, pp. 341-355.
[21] M. Tarsi, Decomposition of a Complete Multigraph into Simple Paths: Nonbalanced Handcuffed Designs, J. Combin. Theory, A 34 (1983), 60-70.
[22] T.W. Tillson, A Hamiltonian decomposition of K∗
2m, 2m ≥ 8,J. Combin. Theory Ser.
B 29 (1) (1980) 68-74.
[23] D. B. West(2001), Introduction to graph theory, Upper Saddle River, NJ :Prentice Hall.
