More extreme equitable colorings of decompositions of [formula omitted] and [formula omitted

Abstract

An H-decomposition of a graph G is a partition P of E(G) into blocks, each element of which induces a copy of H. An (s,p)-equitable block-coloring of the H-decomposition is a coloring of the blocks in P with exactly s colors such that each vertex u is incident with blocks colored with exactly p colors, the blocks containing u being shared out as evenly as possible among the p color classes. Early results in the literature consider the existence of (s,p)-equitable block-colorings of K3- and C4-decompositions of G∈{Kv,Kv−F}, focusing on finding the value of χp′(v), the smallest s for which such a coloring exists.More recently the structure of such colorings has been considered, defining the color vector V(E)=(c1(E),c2(E),…,cs(E)) of an (s,p)-equitable block-coloring E of G, arranged in non-decreasing order, where ci(E) is the number of vertices in G incident with a block of color i. The most interesting cases are where χp′(v)>p, finding the range of c1(E) and of cs(E) among all (s,p)-equitable block-colorings E of G being the main aim. The largest c1 can be and the smallest cs can be for C4-decompositions of Kv−F have been previously found. In this paper the value of the remaining two parameters of most interest are found; namely the smallest c1 can be and the largest cs can be.The extreme colorings found in the main results follow from another problem addressed in this paper: finding extreme (s,p)-equitable edge-colorings of Kv.An important facet of the paper is that the powerful proof technique of graph amalgamations is used for the first time to obtain (s,p)-equitable block-colorings.