Nondisconnecting disentanglements of amalgamated 2‐factorizations of complete multipartite graphs

Abstract

In this paper necessary and suf®cient 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 ®nding hamilton decompositions of K aYb À EU for any 2-factor U of K aYb . #

Hamilton decompositions of complete multipartite graphs with any 2‐factor leave

“…Since each layer is either connected, or is a subgraph of a connected subgraph of G Gð2Þ, and since each layer is connected to the layer above and below, it follows that G Gð2Þ is connected. Therefore the graph G colored with c c satifies properties (1)(2)(3)(4)(5), so the result follows from Theorem 2.1. …”

“…By Theorem 2.1, it suffices to construct an edge-colored graph G that satisfies conditions (1)(2)(3)(4)(5). G is constructed by first amalgamating K, an edge-colored graph isomorphic to K We begin with the edge-coloring c : EðKÞ !…”

“…Recently, Rodger and Leach [5] solved the corresponding problem for the complete bipartite graph K a;b À EðUÞ. In this paper, sufficient conditions are found for the existence of a hamilton decomposition of the complete p-partite graph K m;...;m À EðUÞ where U is a 2-factor of K ðpÞ m that is specified by its cycle lengths.…”

Fair Hamilton Decompositions of Complete Multipartite Graphs

“…Amalgamations have also been used by Buchanan [2] for cracking another difficult problem that has stumped people for years, namely to show that regardless of which 2-factor F is removed from K 2n+1 , the resulting graph K 2n+1 − E(F) has a hamilton decomposition. This result was recently extended to complete bipartite graphs [9], and although progress has been made, the corresponding result for K (p) m still remains open. It is a common objective in mathematics when dealing with partitioning problems to ask for certain characteristics to be evenly shared among the parts on the partition.…”

Connectivity and Colorings of Graphs

“…(The color classes in the given K v may be highly disconnected.) This requires disentangling the amalgamated graph without disconnecting any color class, a technique that is also discussed in [28]. Together, these methods enabled us to solve a problem that seems impossible to obtain using the "tried and tested" techniques; namely, to show when there exists a hamilton decomposition of Kn t n once the edges of any specified 2-factor have been removed.…”