Amalgamations of connected k-factorizations

Amalgamations of factorizations of complete graphs

“…In the standard proof (see, for example, [4,7,11]) the outline graph is "disentangled" by considering in turn each vertex v with f H (v) > 1. A new graph is obtained by splitting v into two vertices v 1 and…”

“…At the end of the next section we shall give a proof using amalgamations. Many combinatorial problems have been solved using amalgamations; see, for example, [1][2][3][4]6,7,11]. Let us sketch how we will use the technique on (t, K, L)-factorizations of K n .…”

“…This kind of outline/amalgamation result is a staple of papers on combinatorial amalgamations, but we were not able to apply the standard techniques (such as those used on problems on amalgamations of factorizations of graphs in [4,6,7,11]). An innovation of this paper is to show how a new technique for finding factorizations of graphs introduced by Hilton and Johnson [5] can be applied to amalgamations.…”

Almost regular edge colorings and regular decompositions of complete graphs

“…The case k = 1 is equivalent to the unipotent case of Cruse's Theorem [8] on embeddings of partial symmetric Latin squares (a proper edge coloring of K 2m with 2m − 1 colors is equivalent to a unipotent symmetric Latin square of order 2m). Using the method of amalgamations, Theorem 4.1 has been extended to obtain necessary and sufficient conditions in the cases where the k-factors are required to be connected [11], or 2-edge-connected [12]. In either case, for k = 2 one obtains Hamilton cycle decompositions of K n (n odd).…”

Connectivity and Colorings of Graphs

“…It enabled us to solve the problem of deciding when an edge-colored multigraph with loops could be the amalgamation of an edge-colored complete graph in which each color class is connected [38]. This allowed us to address the captivating question: When can an edge-colored K v be embedded as an edge-colored K n in which each color class is a connected kfactor?…”