ON THE ASCENDING SUBGRAPH DECOMPOSITIONS OF REGULAR GRAPHS

The definition of the ascending suhgraph decomposition was given by Alavi. It has been conjectured that every graph of positive size has an ascending subgraph decomposition. In this paper it is proved that the regular graphs under some conditions do have an ascending subgraph decomposition.     

On factors of 4‐connected claw‐free graphs*

We consider the existence of several different kinds of factors in 4‐connected claw‐free graphs. This is motivated by the following two conjectures which are in fact equivalent by a recent result of the third author. Conjecture 1 (Thomassen): Every 4‐connected line graph is hamiltonian, i.e., has a connected 2‐factor. Conjecture 2 (Matthews and Sumner): Every 4‐connected claw‐free graph is hamiltonian. We first show that Conjecture 2 is true within the class of hourglass‐free graphs, i.e., graphs that do not contain an induced subgraph isomorphic to two triangles meeting in exactly one vertex. Next we show that a weaker form of Conjecture 2 is true, in which the conclusion is replaced by the conclusion that there exists a connected spanning subgraph in which each vertex has degree two or four. Finally we show that Conjectures 1 and 2 are equivalent to seemingly weaker conjectures in which the conclusion is replaced by the conclusion that there exists a spanning subgraph consisting of a bounded number of paths © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 125–136, 2001     

Clique‐inverse graphs of K3‐free and K4‐free graphs

The clique graph K(G) of a given graph G is the intersection graph of the collection of maximal cliques of G. Given a family ℱ of graphs, the clique‐inverse graphs of ℱ are the graphs whose clique graphs belong to ℱ. In this work, we describe characterizations for clique‐inverse graphs of K₃‐free and K₄‐free graphs. The characterizations are formulated in terms of forbidden induced subgraphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 257–272, 2000     

On transitive decompositions of disconnected graphs

We investigate transitive decompositions of disconnected graphs, and show that these behave very differently from a related class of algebraic graph decompositions, known as homogeneous factorisations. We conclude that although the study of homogeneous factorisations admits a natural reduction to those cases where the graph is connected, the study of transitive decompositions does not.     

On even cycle decompositions of line graphs of cubic graphs

An even cycle decomposition of a graph is a partition of its edges into cycles of even length. In 2012, Markström conjectured that the line graph of every 2-connected cubic graph has an even cycle decomposition and proved this conjecture for cubic graphs with oddness at most 2. However, for 2-connected cubic graphs with oddness 2, Markström only considered these graphs with a chordless 2-factor. (A chordless 2-factor of a graph is a 2-factor consisting of only induced cycles.) In this paper, we first construct an infinite family of 2-connected cubic graphs with oddness 2 and without chordless 2-factors. We then give a complete proof of Markström’s result and further prove this conjecture for cubic graphs with oddness 4.     

Resolvable gregarious cycle decompositions of complete equipartite graphs

The complete multipartite graph K_n(m) with n parts of size m is shown to have a decomposition into n-cycles in such a way that each cycle meets each part of K_n(m); that is, each cycle is said to be gregarious. Furthermore, gregarious decompositions are given which are also resolvable.     

Path and cycle decompositions of complete equipartite graphs: Four parts

We show that a complete equipartite graph with four partite sets has an edge-disjoint decomposition into cycles of length k if and only if k≥3, the partite set size is even, k divides the number of edges in the equipartite graph and the total number of vertices in the graph is at least k. We also show that a complete equipartite graph with four even partite sets has an edge-disjoint decomposition into paths with k edges if and only if k divides the number of edges in the equipartite graph and the total number of vertices in the graph is at least k+1.     

Path and cycle decompositions of complete equipartite graphs: 3 and 5 parts

In 1998 Cavenagh [N.J. Cavenagh, Decompositions of complete tripartite graphs into k-cycles, Australas. J. Combin. 18 (1998) 193-200] gave necessary and sufficient conditions for the existence of an edge-disjoint decomposition of a complete equipartite graph with three parts, into cycles of some fixed length k. Here we extend this to paths, and show that such a complete equipartite graph with three partite sets of size m, has an edge-disjoint decomposition into paths of length k if and only if k divides 3m² and k<3m. Further, extending to five partite sets, we show that a complete equipartite graph with five partite sets of size m has an edge-disjoint decomposition into cycles (and also into paths) of length k with k?3 if and only if k divides 10m² and k?5m for cycles (or k<5m for paths).     

A characterization of P4‐indifference graphs

A graph is a P₄‐indifference graph if it admits a linear ordering ≺ on its vertices such that every chordless path with vertices a, b, c, d and edges ab, bc, cd has either a ≺ b ≺ c ≺ d or d ≺ c ≺ b ≺ a. P₄‐indifference graphs generalize indifference graphs and are perfectly orderable. We give a characterization of P₄‐indifference graphs by forbidden induced subgraphs. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 155‐162, 1999     

Classification of reconfiguration graphs of shortest path graphs with no induced 4-cycles

For any graph $G$ with $a,b\in V\left(G\right)$, a shortest path reconfiguration graph can be formed with respect to $a$ and $b$; we denote such a graph as $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths from $a$ to $b$ in $G$ while two vertices $U,W$ in $V\left(S(G,a,b)\right)$ are adjacent if and only if the vertex sets of the paths that represent $U$ and $W$ differ in exactly one vertex. In a recent paper (Asplund et al., 2018), it was shown that shortest path graphs with girth five or greater are exactly disjoint unions of even cycles and paths. In this paper, we extend this result by classifying all shortest path graphs with no induced 4-cycles.     

One‐factorizations of complete graphs with vertex‐regular automorphism groups

We consider one‐factorizations of K_2n possessing an automorphism group acting regularly (sharply transitively) on vertices. We present some upper bounds on the number of one‐factors which are fixed by the group; further information is obtained when equality holds in these bounds. The case where the group is dihedral is studied in some detail, with some non‐existence statements in case the number of fixed one‐factors is as large as possible. Constructions both for dihedral groups and for some classes of abelian groups are given. © 2002 John Wiley & Sons, Inc. J Combin Designs 10: 1–16, 2002     

图的{P4}——分解

一个图G的路分解是指一路集合使得G的每条边恰好出现在其中一条路上．记Pl长度为l-1的路,如果G能够分解成若干个Pl,则称G存在{Pl}——分解,关于图的给定长路分解问题主要结果有：（i）连通图G存在{P3}-分解当且仅当G有偶数条边（见[1]）;（ii）连通图G存在{P3,P4}-分解当且仅当G不是C3和奇树,这里C3的长度为3的圈而奇树是所有顶点皆度数为奇数的树（见[3]）．本文讨论了3正则图的{P4}--分解情况,并构造证明了边数为3k（k∈Z且k≥2）的完全图Kn和完全二部图Kr,s存在{P4}-分解.     

Hamiltonian decompositions of complete regular s-partite graphs

In this paper we give a procedure by which Hamiltonian decompositions of the s-partite graph K_n,n,…,n, where (s − 1)n is even, can be constructed. For 2t⩽s, 1⩽a₁⩽…⩽a_t⩽n, we find conditions which are necessary and sufficient for a decomposition of the edge-set of K_a1a2…,a_t into (s − 1)n/2 classes, each class consisting of disjoint paths, to be extendible to a Hamiltonian decomposition of the complete s-partite graph K_rmn,n,…,n so that each of the classes forms part of a Hamiltonian cycle.     

图的{P4}-分解

一个图G的路分解是指一路集合使得G的每条边恰好出现在其中一条路上.记Pl长度为l-1的路,如果G能够分解成若干个Pl,则称G存在{Pl}—分解.关于图的给定长路分解问题主要结果有:(i)连通图G存在{P3}—分解当且仅当G有偶数条边(见[1]);(ii)连通图G存在{P3,P4}—分解当且仅当G不是C3和奇树,这里C3的长度为3的圈而奇树是所有顶点皆度数为奇数的树(见[3]).本文讨论了3正则图的{P4}—分解情况,并构造证明了边数为3k(k热∈Z且k≥2)的完全图Kn和完全二部图Kr,s存在{P4}—分解.     

On 4-chromatic edge-critical regular graphs of high connectivity

New examples of 4-chromatic edge-critical r-regular and r-connected graphs are presented for r = 6,8,10.     

The number of Hamiltonian decompositions of regular graphs

A Hamilton cycle in a graph Γ is a cycle passing through every vertex of Γ. A Hamiltonian decomposition of Γ is a partition of its edge set into disjoint Hamilton cycles. One of the oldest results in graph theory is Walecki’s theorem from the 19th century, showing that a complete graph K _n on an odd number of vertices n has a Hamiltonian decomposition. This result was recently greatly extended by Kühn and Osthus. They proved that every r-regular n-vertex graph Γ with even degree r = cn for some fixed c > 1/2 has a Hamiltonian decomposition, provided n = n(c) is sufficiently large. In this paper we address the natural question of estimating H(Γ), the number of such decompositions of Γ. Our main result is that H(Γ) = r ^(1+o(1))nr/2. In particular, the number of Hamiltonian decompositions of K _n is \({n^{\left( {1 + o\left( 1 \right)} \right){n^2}/2}}\).     

Acyclic 4‐Choosability of Planar Graphs with No 4‐ and 5‐Cycles

Every planar graph is known to be acyclically 7‐choosable and is conjectured to be acyclically 5‐choosable (O. V. Borodin, D. G. Fon‐Der‐Flaass, A. V. Kostochka, E. Sopena, J Graph Theory 40 (2002), 83–90). This conjecture if proved would imply both Borodin's (Discrete Math 25 (1979), 211–236) acyclic 5‐color theorem and Thomassen's (J Combin Theory Ser B 62 (1994), 180–181) 5‐choosability theorem. However, as yet it has been verified only for several restricted classes of graphs. Some sufficient conditions are also obtained for a planar graph to be acyclically 4‐ and 3‐choosable. In particular, the acyclic 4‐choosability was proved for the following planar graphs: without 3‐, 4‐, and 5‐cycles (M. Montassier, P. Ochem, and A. Raspaud, J Graph Theory 51 (2006), 281–300), without 4‐, 5‐, and 6‐cycles, or without 4‐, 5‐, and 7‐cycles, or without 4‐, 5‐, and intersecting 3‐cycles (M. Montassier, A. Raspaud, W. Wang, Topics Discrete Math (2006), 473–491), and neither 4‐ and 5‐cycles nor 8‐cycles having a triangular chord (M. Chen and A. Raspaud, Discrete Math. 310(15–16) (2010), 2113–2118). The purpose of this paper is to strengthen these results by proving that each planar graph without 4‐ and 5‐cycles is acyclically 4‐choosable.     

Roman domination in regular graphs

A Roman domination function on a graph G=(V(G),E(G)) is a function f:V(G)→{0,1,2} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. The weight of a Roman dominating function is the value f(V(G))=∑_u∈V(G)f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. Cockayne et al. [E. J. Cockayne et al. Roman domination in graphs, Discrete Mathematics 278 (2004) 11-22] showed that γ(G)≤γ_R(G)≤2γ(G) and defined a graph G to be Roman if γ_R(G)=2γ(G). In this article, the authors gave several classes of Roman graphs: P_3k,P_3k+2,C_3k,C_3k+2 for k≥1, K_m,n for min{m,n}≠2, and any graph G with γ(G)=1; In this paper, we research on regular Roman graphs and prove that: (1) the circulant graphs and , n⁄≡1 (mod (2k+1)), (n≠2k) are Roman graphs, (2) the generalized Petersen graphs P(n,2k+1)( (mod 4) and ), P(n,1) (n⁄≡2 (mod 4)), P(n,3) ( (mod 4)) and P(11,3) are Roman graphs, and (3) the Cartesian product graphs are Roman graphs.