Face 2‐Colorable Embeddings with Faces of Specified Lengths

Suppose and are arbitrary lists of positive integers. In this article, we determine necessary and sufficient conditions on M and N for the existence of a simple graph G, which admits a face 2‐colorable planar embedding in which the faces of one color have boundary lengths and the faces of the other color have boundary lengths . Such a graph is said to have a planar ‐biembedding. We also determine necessary and sufficient conditions on M and N for the existence of a simple graph G whose edge set can be partitioned into r cycles of lengths and also into t cycles of lengths . Such a graph is said to be ‐decomposable.     

Decomposing Graphs of High Minimum Degree into 4‐Cycles

If a graph G decomposes into edge‐disjoint 4‐cycles, then each vertex of G has even degree and 4 divides the number of edges in G. It is shown that these obvious necessary conditions are also sufficient when G is any simple graph having minimum degree at least , where n is the number of vertices in G. This improves the bound given by Gustavsson (PhD Thesis, University of Stockholm, 1991), who showed (as part of a more general result) sufficiency for simple graphs with minimum degree at least . On the other hand, it is known that for arbitrarily large values of n there exist simple graphs satisfying the obvious necessary conditions, having n vertices and minimum degree , but having no decomposition into edge‐disjoint 4‐cycles. We also show that if G is a bipartite simple graph with n vertices in each part, then the obvious necessary conditions for G to decompose into 4‐cycles are sufficient when G has minimum degree at least .     

Decompositions of Graphs into Fans and Single Edges

Given two graphs G and H , an H‐decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a graph isomorphic to H . Let be the smallest number ? such that any graph G of order n admits an H‐decomposition with at most ? parts. Pikhurko and Sousa conjectured that for and all sufficiently large n , where denotes the maximum number of edges in a graph on n vertices not containing H as a subgraph. Their conjecture has been verified by Özkahya and Person for all edge‐critical graphs H . In this article, the conjecture is verified for the k‐fan graph. The k‐fan graph , denoted by , is the graph on vertices consisting of k triangles that intersect in exactly one common vertex called the center of the k‐fan.     

On Rooted Packings,Decompositions, and Factors of Graphs

In this article, we study so‐called rooted packings of rooted graphs. This concept is a mutual generalization of the concepts of a vertex packing and an edge packing of a graph. A rooted graph is a pair , where G is a graph and . Two rooted graphs and are isomorphic if there is an isomorphism of the graphs G and H such that S is the image of T in this isomorphism. A rooted graph is a rooted subgraph of a rooted graph if H is a subgraph of G and . By a rooted ‐packing into a rooted graph we mean a collection of rooted subgraphs of isomorphic to such that the sets of edges are pairwise disjoint and the sets are pairwise disjoint. In this article, we concentrate on studying maximum ‐packings when H is a star. We give a complete classification with respect to the computational complexity status of the problems of finding a maximum ‐packing of a rooted graph when H is a star. The most interesting polynomial case is the case when H is the 2‐edge star and S contains the center of the star only. We prove a min–max theorem for ‐packings in this case.     

完全图与完全n部图的完全多部图分解

§ 1　IntroductionLet V(G) and E(G) be the vertex setand the edge setof a graph G,respectively.Fori=1 ,...,p,if V(Gi) V(G) ,E(Gi)∩ E(Gj) = for i≠ j,and∪pi=1 E(Gi) =E(G) ,then wecall{ G1 ,...,GP} a decomposition of G.Let[i,j] be the integer interval including i and j.Let Knbe a complete graph with the vertex set[1 ,n] .For m disjointsubsets A1 ,...Amof[1 ,n] ,let K(A1 ,...,Am) be a complete m-partite graph having partite-sets A1 ,...,Am.If| Ai| =1 ,Ai is called a S-set;otherwi…     

Hamilton cycle decomposition of 6‐regular circulants of odd order

The circulant G = C(n,S), where , is the graph with vertex set Z_n and edge set . It is shown that for n odd, every 6‐regular connected circulant C(n, S) is decomposable into Hamilton cycles. © 2006 Wiley Periodicals, Inc. J Combin Designs     

Almost regular edge colorings and regular decompositions of complete graphs

For k = 1 and k = 2, we prove that the obvious necessary numerical conditions for packing t pairwise edge‐disjoint k‐regular subgraphs of specified orders m₁,m₂,… ,m_t in the complete graph of order n are also sufficient. To do so, we present an edge‐coloring technique which also yields new proofs of various known results on graph factorizations. For example, a new construction for Hamilton cycle decompositions of complete graphs is given. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 499–506, 2008     

Decomposition of Complete Graphs into Isomorphic Complete Bipartite Graphs

A decomposition of a complete graph into disjoint copies of a complete bipartite graph is called a ‐design of order n. The existence problem of ‐designs has been completely solved for the graphs for , for , K_{2, 3} and K_{3, 3}. In this paper, I prove that for all , if there exists a ‐design of order N, then there exists a ‐design of order n for all (mod ) and . Giving necessary direct constructions, I provide an almost complete solution for the existence problem for complete bipartite graphs with fewer than 18 edges, leaving five orders in total unsolved.     

Vertex disjoint equivalent subgraphs of order 3

Let k be a fixed integer at least 3. It is proved that every graph of order (2k ? 1 ? 1/k)n + O(1) contains n vertex disjoint induced subgraphs of order k such that these subgraphs are equivalent to each other and they are equivalent to one of four graphs: a clique, an independent set, a star, or the complement of a star. In particular, by substituting 3 for k, it is proved that every graph of order 14n/3 + O(1) contains n vertex disjoint induced subgraphs of order 3 such that they are equivalent to each other. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 159–166, 2007     

Hamilton cycle rich two‐factorizations of complete graphs

For all integers n ≥ 5, it is shown that the graph obtained from the n‐cycle by joining vertices at distance 2 has a 2‐factorization is which one 2‐factor is a Hamilton cycle, and the other is isomorphic to any given 2‐regular graph of order n. This result is used to prove several results on 2‐factorizations of the complete graph K_n of order n. For example, it is shown that for all odd n ≥ 11, K_n has a 2‐factorization in which three of the 2‐factors are isomorphic to any three given 2‐regular graphs of order n, and the remaining 2‐factors are Hamilton cycles. For any two given 2‐regular graphs of even order n, the corresponding result is proved for the graph K_n ‐ I obtained from the complete graph by removing the edges of a 1‐factor. © 2004 Wiley Periodicals, Inc.     

On Hamilton decompositions of infinite circulant graphs

The natural infinite analog of a (finite) Hamilton cycle is a two‐way‐infinite Hamilton path (connected spanning 2‐valent subgraph). Although it is known that every connected 2k‐valent infinite circulant graph has a two‐way‐infinite Hamilton path, there exist many such graphs that do not have a decomposition into k edge‐disjoint two‐way‐infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2k‐valent connected circulant graph has a decomposition into k edge‐disjoint Hamilton cycles. We settle the problem of decomposing 2k‐valent infinite circulant graphs into k edge‐disjoint two‐way‐infinite Hamilton paths for , in many cases when , and in many other cases including where the connection set is or .     

Bipartite 2‐Factorizations of Complete Multipartite Graphs

It is shown that if K is any regular complete multipartite graph of even degree, and F is any bipartite 2‐factor of K, then there exists a factorization of K into F; except that there is no factorization of K_{6, 6} into F when F is the union of two disjoint 6‐cycles.     

Decomposing complete equipartite graphs into short even cycles

In this article we find necessary and sufficient conditions to decompose a complete equipartite graph into cycles of uniform length, in the case that the length is both even and short relative to the number of parts. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:131‐143, 2011     

Two Hamiltonian cycles

If the line graph of a graph G decomposes into Hamiltonian cycles, what is G? We answer this question for decomposition into two cycles.     

Regular path decompositions of odd regular graphs

Kotzig asked in 1979 what are necessary and sufficient conditions for a d‐regular simple graph to admit a decomposition into paths of length d for odd d>3. For cubic graphs, the existence of a 1‐factor is both necessary and sufficient. Even more, each 1‐factor is extendable to a decomposition of the graph into paths of length 3 where the middle edges of the paths coincide with the 1‐factor. We conjecture that existence of a 1‐factor is indeed a sufficient condition for Kotzig's problem. For general odd regular graphs, most 1‐factors appear to be extendable and we show that for the family of simple 5‐regular graphs with no cycles of length 4, all 1‐factors are extendable. However, for d>3 we found infinite families of d‐regular simple graphs with non‐extendable 1‐factors. Few authors have studied the decompositions of general regular graphs. We present examples and open problems; in particular, we conjecture that in planar 5‐regular graphs all 1‐factors are extendable. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 114–128, 2010     

HAMILTONIAN DECOMPOSITION OF CAYLEY GRAPHS OF ORDERS p~2 AND pq

1. IntroductionLet G be a finite group and S a subset of G such that S--1 ~ S, and 1 f S. The Cayleygraph Cay (G, S) is defined as the simple graph with V ~ G, and E = {glgZ I g,'g, or g,'g,6 S, gi, gi E G}. Cay (G, S) is vertex-transitive, and it is connected if and only if (S) = G,i.e. S is a generating set of G[1]. If G = Zn, then Cay (Zn, S) is called a circulant graph. Ithas been proved that any connected Cayley graph on a finite abelian group is hamiltonianl2].Furthermore, …     

关于整数分拆的Alavi猜想的证明

Ｙ．Ａｌａｖｉ，Ｐ．Ｅｒｄｓ等人在［１］中提出猜想：设自然数α＿１，α＿２…α＿ｋ满足且，则可以划分成ｋ个互不相交子集Ｓ＿１，Ｓ＿２，···，Ｓ＿ｋ，满足．本文证明了这个猜想。