Some equitably 3-colourable cycle decompositions of complete equipartite graphs

Let G be a graph in which each vertex has been coloured using one of k colours, say c(1), c(2),.. , c(k). If an m-cycle C in G has n(i) vertices coloured c(i), i = 1, 2,..., k, and vertical bar n(i) - n(j)vertical bar <= 1 for any i, j is an element of {1, 2,..., k}, then C is said to be equitably k-coloured. An m-cycle decomposition C of a graph G is equitably k-colourable if the vertices of G can be coloured so that every m-cycle in W is equitably k-coloured. For m = 3, 4 and 5 we completely settle the existence question for equitably 3-colourable m-cycle decompositions of complete equipartite graphs. (c) 2005 Elsevier B.V. All rights reserved.

Decompositions of complete tripartite graphs into triangles with an edge attached

Let K(r, s, t) denote the complete tripartite graph with partite sets of size r, s and t, where r less than or equal to s less than or equal to t. Let D be the graph consisting of a triangle with an edge attached. We show that K(r, s, t) may be decomposed into copies of D if and only if 4 divides rs + st + rt and t less than or equal to 3rs/(r + s).

Steiner trade spectra of complete partite graphs

The Steiner trade spectrum of a simple graph G is the set of all integers t for which there is a simple graph H whose edges can be partitioned into t copies of G in two entirely different ways. The Steiner trade spectra of complete partite graphs were determined in all but a few cases in a recent paper by Billington and Hoffman (Discrete Math. 250 (2002) 23). In this paper we resolve the remaining cases. (C) 2004 Elsevier B.V. All rights reserved.

Packing lambda-fold complete multipartite graphs with 4-cycles

A maximum packing of any lambda-fold complete multipartite graph (where there are lambda edges between any two vertices in different parts) with edge-disjoint 4- cycles is obtained and the size of each minimum leave is given. Moreover, when lambda =2, maximum 4-cycle packings are found for all possible leaves.

Common multiples of complete graphs and a 4-cycle

A graph G is a common multiple of two graphs H-1 and H-2 if there exists a decomposition of G into edge-disjoint copies of H-1 and also a decomposition of G into edge-disjoint copies of H-2. In this paper, we consider the case where H-1 is the 4-cycle C-4 and H-2 is the complete graph with n vertices K-n. We determine, for all positive integers n, the set of integers q for which there exists a common multiple of C-4 and K-n having precisely q edges. (C) 2003 Elsevier B.V. All rights reserved.

Cube factorizations of complete graphs

A cube factorization of the complete graph on n vertices, K-n, is a 3-factorization of & in which the components of each factor are cubes. We show that there exists a cube factorization of & if and only if n equivalent to 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. (C) 2004 Wiley Periodicals, Inc.

Decompositions of complete graphs into triangles and Hamilton cycles

For all odd integers n greater than or equal to 1, let G(n) denote the complete graph of order n, and for all even integers n greater than or equal to 2 let G,, denote the complete graph of order n with the edges of a 1-factor removed. It is shown that for all non-negative integers h and t and all positive integers n, G, can be decomposed into h Hamilton cycles and t triangles if and only if nh + 3t is the number of edges in G(n). (C) 2004 Wiley Periodicals, Inc.

Decomposition of complete graphs into 5-cubes

Necessary conditions for the complete graph on n vertices to have a decomposition into 5-cubes are that 5 divides it - 1 and 80 divides it (it - 1)/2. These are known to be sufficient when n is odd. We prove them also sufficient for it even, thus completing the spectrum problem for the 5-cube and lending further weight to a long-standing conjecture of Kotzig. (c) 2005 Wiley Periodicals, Inc.