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…     

Factorizations of complete multipartite graphs into generalized cubes

For a positive integer d, the usual d‐dimensional cube Q_d is defined to be the graph (K₂)^d, the Cartesian product of d copies of K₂. We define the generalized cube Q(K_k, d) to be the graph (K_k)^d for positive integers d and k. We investigate the decomposition of the complete multipartite graph K into factors that are vertex‐disjoint unions of generalized cubes Q(K_k, d_i), where k is a power of a prime, n and j are positive integers with j ≤ n, and the d_i may be different in different factors. We also use these results to partially settle a problem of Kotzig on Q_d‐factorizations of K_n. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 144–150, 2000     

The inducibility of complete bipartite graphs

For any s ≥ 1 and t ≥ (^S₂), we prove that among all graphs with n vertices the graph that contains the maximal number of induced copies of K_{t, t+s} for any fixed s ≥ 1 and t ≥ (^s₂) is K_{(n/2)+α(n/2)-α} for some function α = o(n). We show that this is not valid for t < (^s₂). Analogous results for complete multipartite graphs are also obtained.     

Symmetric Hamilton cycle decompositions of complete graphs minus a 1‐factor

Let n≥2 be an integer. The complete graph K_n with a 1‐factor F removed has a decomposition into Hamilton cycles if and only if n is even. We show that K_n−F has a decomposition into Hamilton cycles which are symmetric with respect to the 1‐factor F if and only if n≡2, 4 mod 8. We also show that the complete bipartite graph K_{n, n} has a symmetric Hamilton cycle decomposition if and only if n is even, and that if F is a 1‐factor of K_{n, n}, then K_{n, n}−F has a symmetric Hamilton cycle decomposition if and only if n is odd. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:1‐15, 2010     

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.     

Self-dual embeddings of complete multipartite graphs

In this paper we examine self-dual embeddings of complete multipartite graphs, focusing primarily on K_m(n) having m parts each of size n. If m = 2, then n must be even. If the embedding is on an orientable surface, then an Euler characteristic argument shows that no such embedding exists when n is odd and m ? 2, 3 (mod 4); there is no such restriction for embeddings on nonorientable surfaces. We show that these embeddings exist with a few small exceptions. As a corollary, every group has a Cayley graph with a self-dual embedding. Our main technique is an addition construction that combines self-dual embeddings of two subgraphs into a self-dual embedding of their union. We also apply this technique to nonregular multipartite graphs and to cubes.     

P 3-factorization of complete multipartite graphs

In this note it is shown that a necessary and sufficient condition for the existence of a P3-factorizatlon of complete multipartite graph λK, is (1) m≥3, (2) mn≡0(mod 3) and (3)λ(m-1)n≡0(mod 4).     

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 4-semiregular 1-factorizations of complete graphs and complete bipartite graphs

A 4-semiregular 1-factorization is a 1-factorization in which every pair of distinct 1-factors forms a union of 4-cycles. LetK be the complete graphK _2nor the complete bipartite graphK _{n, n} .We prove that there is a 4-semiregular 1-factorization ofK if and only ifn is a power of 2 andn2, and 4-semiregular 1-factorizations ofK are isomorphic, and then we determine the symmetry groups. They are known for the case of the complete graphK _2n ,however, we prove them in a different method.     

On the decomposition of Kn into complete m-partite graphs

Graham and Pollak [3] proved that n ?1 is the minimum number of edge-disjoint complete bipartite subgraphs into which the edges of K_n can be decomposed. Using a linear algebraic technique, Tverberg [2] gives a different proof of that result. We apply his technique to show that for “almost all n,” ? (n + m ?3)/(m ?1) ? is the minimum number of edge-disjoint complete m-partite subgraphs in a decomposition of K_n.     

Multicolored trees in complete graphs

A multicolored tree is a tree whose edges have different colors. Brualdi and Hollingsworth 5 proved in any proper edge coloring of the complete graph K_2n(n > 2) with 2n ? 1 colors, there are two edge‐disjoint multicolored spanning trees. In this paper we generalize this result showing that if (a₁,…, a_k) is a color distribution for the complete graph K_n, n ≥ 5, such that , then there exist two edge‐disjoint multicolored spanning trees. Moreover, we prove that for any edge coloring of the complete graph K_n with the above distribution if T is a non‐star multicolored spanning tree of K_n, then there exists a multicolored spanning tree T' of K_n such that T and T' are edge‐disjoint. Also it is shown that if K_n, n ≥ 6, is edge colored with k colors and , then there exist two edge‐disjoint multicolored spanning trees. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 221–232, 2007     

Cube factorizations of complete graphs

A cube factorization of the complete graph on n vertices, K_n, is a 3‐factorization of K_n in which the components of each factor are cubes. We show that there exists a cube factorization of K_n if and only if n ≡ 16 (mod 24), thus providing a new family of uniform 3‐factorizations as well as a partial solution to an open problem posed by Kotzig in 1979. © 2004 Wiley Periodicals, Inc.     

Regular orientable embeddings of complete bipartite graphs

In this paper, it will be shown that the isomorphism classes of regular orientable embeddings of the complete bipartite graph K_n,n are in one‐to‐one correspondence with the permutations on n elements satisfying a given criterion, and the isomorphism classes of them are completely classified when n is a product of any two (not necessarily distinct) prime numbers. For other n, a lower bound of the number of those isomorphism classes of K_n,n is obtained. As a result, many new regular orientable embeddings of the complete bipartite graph are constructed giving an answer of Nedela‐?koviera's question raised in 12 . © 2005 Wiley Periodicals, Inc. J Graph Theory     

Strong difference families over arbitrary graphs

The concept of a strong difference family formally introduced in Buratti [J Combin Designs 7 (1999), 406–425] with the aim of getting group divisible designs with an automorphism group acting regularly on the points, is here extended for getting, more generally, sharply‐vertex‐transitive Γ‐decompositions of a complete multipartite graph for several kinds of graphs Γ. We show, for instance, that if Γ has e edges, then it is often possible to get a sharply‐vertex‐transitive Γ‐decomposition of K_{m × e} for any integer m whose prime factors are not smaller than the chromatic number of Γ. This is proved to be true whenever Γ admits an α‐labeling and, also, when Γ is an odd cycle or the Petersen graph or the prism T₅ or the wheel W₆. We also show that sometimes strong difference families lead to regular Γ‐decompositions of a complete graph. We construct, for instance, a regular cube‐decomposition of K_16m for any integer m whose prime factors are all congruent to 1 modulo 6. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 443–461, 2008     

Line graphs of complete multipartite graphs have small cycle double covers

It has been shown by MacGillivray and Seyffarth (Austral. J. Combin. 24 (2001) 91) that bridgeless line graphs of complete graphs, complete bipartite graphs, and planar graphs have small cycle double covers. In this paper, we extend the result for complete bipartite graphs, and show that the line graph of any complete multipartite graph (other than K_1,2) has a small cycle double cover.     

Properly colored hamilton cycles in edge-colored complete graphs

It is shown that, for ϵ>0 and n>n₀(ϵ), any complete graph K on n vertices whose edges are colored so that no vertex is incident with more than (1-1/\sqrt2-\epsilon)n edges of the same color contains a Hamilton cycle in which adjacent edges have distinct colors. Moreover, for every k between 3 and n any such K contains a cycle of length k in which adjacent edges have distinct colors. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 , 179–186 (1997)     

Nondisconnecting disentanglements of amalgamated 2‐factorizations of complete multipartite graphs

In this paper necessary and sufficient conditions are found for an edge‐colored graph H to be the homomorphic image of a 2‐factorization of a complete multipartite graph G in which each 2‐factor of G has the same number of components as its corresponding color class in H. This result is used to completely solve the problem of finding hamilton decompositions of K_a,b ? E(U) for any 2‐factor U of K_a,b. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 460–467, 2001     

Isomorphic decomposition of complete graphs into linear forests

The well-known theorem of Kirkman states that every complete graph K_2n of order 2n is 1-factorable or, equivalently, is nK₂-decomposable. This result is generalized to any linear forest of size n without isolated vertices.     

Kp,q-factorization of complete bipartite graphs

Let K _m,nbe a complete bipartite graph with two partite sets having m and n vertices, respectively. A K _p,q-factorization of K _m,n is a set of edge-disjoint K _p,q-factors of K _m,n which partition the set of edges of K _m,n. When p = 1 and q is a prime number, Wang, in his paper “On K _1,k -factorizations of a complete bipartite graph” (Discrete Math, 1994, 126: 359—364), investigated the K _1,q -factorization of K _m,nand gave a sufficient condition for such a factorization to exist. In the paper “K _1,k -factorizations of complete bipartite graphs” (Discrete Math, 2002, 259: 301—306), Du and Wang extended Wang’s result to the case that q is any positive integer. In this paper, we give a sufficient condition for K _m,n to have a K _p,q-factorization. As a special case, it is shown that the Martin’s BAC conjecture is true when p : q = k : (k+ 1) for any positive integer k.     

On choosability of some complete multipartite graphs and Ohba's conjecture

A graph G is said to be chromatic-choosable if ch(G)=χ(G). Ohba has conjectured that every graph G with 2χ(G)+1 or fewer vertices is chromatic-choosable. It is clear that Ohba's conjecture is true if and only if it is true for complete multipartite graphs. But for complete multipartite graphs, the graphs for which Ohba's conjecture has been verified are nothing more than K_{3*2,2*(k-3),1}, K_3,2*(k-1), and K_{s+3,2*(k-s-1),1*s}. These results have been obtained indirectly from the investigation about complete multipartite graphs by Gravier and Maffray and by Enomoto et al. In this paper we show that Ohba's conjecture is true for complete multipartite graphs K_{4,3,2*(k-4),1*2} and K_{5,3,2*(k-5),1*3}. By the way, we give some discussions about a result of Enomoto et al.