Decompositions of Complete Graphs

If s₁, s₂, ..., s_t are integers such that n – 1 = s₁ +s₂ + ... + s_t and such that for each i (1 i t), 2 s_i n –1 and s_in is even, then K_n can be expressed as the union G₁G₂...G_tof t edge-disjoint factors, where for each i, G_i is s_i-regularand s_i-connected. Moreover, whenever s_i = s_j, G_i and G_j areisomorphic. 1991 Mathematics Subject Classification 05C70.     

Induced Decompositions of Graphs

We consider those graphs G that admit decompositions into copies of a fixed graph F, each copy being an induced subgraph of G. We are interested in finding the extremal graphs with this property, that is, those graphs G on n vertices with the maximum possible number of edges. We discuss the cases where F is a complete equipartite graph, a cycle, a star, or a graph on at most four vertices.     

Chromatically Supremal Decompositions of Graphs

If G is a graph, a G-decomposition of a host graph H is a partition of the edges of H into subgraphs of H which are isomorphic to G. The chromatic index of a G-decomposition of H is the minimum number of colors required to color the parts of the decomposition so that parts which share a common node get different colors. We establish an upper bound on the chromatic index and characterize those decompositions which achieve it. The structurally most interesting of the decompositions with maximal chromatic index are associated with (v, k, 1)-designs.     

On Cyclic Decompositions of Complete Graphs into Tripartite Graphs

We introduce two new labelings for tripartite graphs and show that if a graph G with n edges admits either of these labelings, then there exists a cyclic G‐decomposition of for every positive integer x. We also show that if G is the union of two vertext‐disjoint cycles of odd length, other than , then G admits one of these labelings.     

Induced Decompositions of Highly Dense Graphs

Given two graphs F and G, an induced F‐decomposition of G is a partition of into induced subgraphs isomorphic to F. Bondy and Szwarcfiter [J. Graph Theory, DOI: 10.1002/jgt.21654] defined the value as the maximum number of edges in a graph of order n admitting an induced F‐decomposition and determined the value of for some graphs (and families of graphs). In this article, we prove that is valid for all graphs F. We also present tighter asymptotic bounds for some of the small graphs for which the exact value of remains unknown. The proofs are based on the heavy use of various classes of Kneser graphs and hypergraphs.     

Ear Decompositions of Matching Covered Graphs

G different from and has at least Δ edge-disjoint removable ears, where Δ is the maximum degree of G. This shows that any matching covered graph G has at least Δ! different ear decompositions, and thus is a generalization of the fundamental theorem of Lovász and Plummer establishing the existence of ear decompositions. We also show that every brick G different from and has Δ− 2 edges, each of which is a removable edge in G, that is, an edge whose deletion from G results in a matching covered graph. This generalizes a well-known theorem of Lovász. We also give a simple proof of another theorem due to Lovász which says that every nonbipartite matching covered graph has a canonical ear decomposition, that is, one in which either the third graph in the sequence is an odd-subdivision of or the fourth graph in the sequence is an odd-subdivision of . Our method in fact shows that every nonbipartite matching covered graph has a canonical ear decomposition which is optimal, that is one which has as few double ears as possible. Most of these results appear in the Ph. D. thesis of the first author [1], written under the supervision of the second author. Received: November 3, 1997     

On Connected Resolving Decompositions in Graphs

For an ordered k-decomposition of a connected graph G and an edge e of G, the -code of e is the k-tuple where d(e, G _i) is the distance from e to G _i. A decomposition is resolving if every two distinct edges of G have distinct -codes. The minimum k for which G has a resolving k-decomposition is its decomposition dimension dim _d (G). A resolving decomposition of G is connected if each G _i is connected for 1 i k. The minimum k for which G has a connected resolving k-decomposition is its connected decomposition number cd(G). Thus 2 dim _d (G) cd(G) m for every connected graph G of size m 2. All nontrivial connected graphs of size m with connected decomposition number 2 or m have been characterized. We present characterizations for connected graphs of size m with connected decomposition number m – 1 or m – 2. It is shown that each pair s, t of rational numbers with 0 < s t 1, there is a connected graph G of size m such that dim _d (G)/m = s and cd(G)/m = t.     

Monochromatic Kr‐Decompositions of Graphs

Given graphs G and H, and a coloring of the edges of G with k colors, a monochromatic 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 monochromatic graph isomorphic to H. Let be the smallest number ? such that any graph G of order n and any coloring of its edges with k colors, admits a monochromatic H‐decomposition with at most ? parts. Here, we study the function for and .     

Petersen Graph Decompositions of Complete Multipartite Graphs

Let P be the Petersen graph, and K _u(h) the complete multipartite graph with u parts of size h. A decomposition of K _u(h) into edge-disjoint copies of the Petersen graph P is called a P-decomposition of K _u(h) or a P-group divisible design of type h ^u . In this paper, we show that there exists a P-decomposition of K _u(h) if and only if h²u(u-1) o 0 mod 30{h^2u(u-1)\equiv 0 \pmod {30}} , h(u-1) o 0 mod 3{h(u-1)\equiv 0\pmod 3} , and u ≥ 3 with a definite exception (h, u) = (1, 10).     

Rank 3 Transitive Decompositions of Complete Multipartite Graphs

A transitive decomposition of a graph is a partition of the edge or arc set giving a set of subgraphs which are preserved and permuted transitively by a group of automorphisms of the graph. This paper deals with transitive decompositions of complete multipartite graphs preserved by an imprimitive rank 3 permutation group. We obtain a near-complete classification of these when the group in question has an almost simple component.     

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.     

有限循环群上的Cayley有向图的哈密顿分解

本文给出了有限循环群上的Cayley有向图Cay(M,G)可哈密顿分解的一个充分条件,并证明了当|M|=2时此条件还是必要的.     

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.     

Decompositions of Complete Multipartite Graphs into Cycles of Even Length

Necessary and sufficient conditions for the existence of an edge-disjoint decomposition of any complete multipartite graph into even length cycles are investigated. Necessary conditions are listed and sufficiency is shown for the cases when the cycle length is 4, 6 or 8. Further results concerning sufficiency, provided certain “small” decompositions exist, are also given for arbitrary even cycle lengths. Revised: November 28, 1997     

Ascending Subgraph Decompositions in Oriented Complete Balanced Tripartite Graphs

In 1987, Alavi, Boals, Chartrand, Erdös, and Oellermann conjectured that all graphs have an ascending subgraph decomposition (ASD). Though different classes of graphs have been shown to have an ASD, the conjecture remains open. In this paper we investigate the similar problem for digraphs. In particular, we will show that any orientation of a compete balanced tripartite graph has an ASD.     

Tree Decompositions of Graphs: Saving Memory in Dynamic Programming

We propose an effective heuristic to save memory in dynamic programming on tree decompositions when solving graph optimization problems. The introduced “anchor technique” is closely related to a tree-like set covering problem.     

6个素数阶完全图的循环图分解

研究素数阶完全图分解为循环图的方法 ,给出计算它的子图的团数的一种算法 ,得到 3个三色 ,3个四色 Ramsey数的新的下界 :R(3,3,13) 194 ,R(3,4 ,11) 2 12 ,R(3,6 ,13) 52 2 ,R(3,3,4 ,10 ) 380 ,R(3,3,6 ,14) 1154,R(3,4 ,5,13) 10 94     

Large Sets of Hamilton Cycle and Path Decompositions of Complete Bipartite Graphs

In this paper, we determine the existence spectrums for large sets of Hamilton cycle and path (resp. directed Hamilton cycle and path) decompositions of ??K _{m, n} (resp. ${\lambda K^{*}_{m,n}}$ ).     

New i‐Perfect Cycle Decompositions via Vertex Colorings of Graphs

Generalizing a result by Buratti et al.[M. Buratti, F. Rania, and F. Zuanni, Some constructions for cyclic perfect cycle systems, Discrete Math 299 (2005), 33–48], we present a construction for i‐perfect k‐cycle decompositions of the complete m‐partite graph with parts of size k. These decompositions are sharply vertex‐transitive under the additive group of with R a suitable ring of order m. The construction works whenever a suitable i‐perfect map exists. We show that for determining the set of all triples for which such a map exists, it is crucial to calculate the chromatic numbers of some auxiliary graphs. We completely determine this set except for one special case where is the product of two distinct primes, is even, and . This result allows us to obtain a plethora of new i‐perfect k‐cycle decompositions of the complete graph of order (mod 2k) with k odd. In particular, if k is a prime, such a decomposition exists for any possible i provided that .