Equitable block-colorings of C4-decompositions of Kv−F

In this paper we consider the existence of (s,p)-equitable block-colorings of 4-cycle decompositions of Kv−F, where F is a 1-factor of Kv. In such colorings, the 4-cycles are colored with s colors in such a way that, for each vertex u, the 4-cycles containing u are colored with p colors so that the number of such 4-cycles of each color is within one of the number of such 4-cycles of each of the other p−1 colors. Of primary interest is settling the values of χp′(v) and χ̄p′(v), namely the least and greatest values of s for which there exists such a block-coloring of some 4-cycle decomposition of Kv−F. In this paper, several general results are established, both existence and non-existence theorems. These are then used to find, for all possible values of v, the values of χp′(v) when p∈{2,3,4} and χ̄2′(v), and to provide good upper bounds on χ̄3′(v).