Extreme equitable block colorings of $C_4$-decompositions of $K_v-F$

A C4-decomposition of Kv−F , where F is a 1-factor of Kv and C4 is the cycle of length 4, is a partition P of E(Kv − F ) into sets, each element of which induces a C4 (called a block). A function assigning a color to each block defined by P is said to be an (s, p)-equitable block-coloring if: exactly s colors are used; each vertex v is incident with blocks colored with exactly p colors; and the blocks containing v are shared out as evenly as possible among the p color classes. Of particular interest is the value of χp(v), the smallest value of s for which there exists an (s, p)-equitable block-coloring of some C4decomposition of Kv − F . In this paper the value of χp(v) is found in the most interesting cases where traditional proof techniques are rendered useless, namely when χp(v) > p. This settles an open problem in a recent paper. Finally, the study of the structure within such equitable block-colorings is developed. In all cases where χp(v) > p, the problems of finding the value of the smallest color class when it is as large as possible, ψ′ 1(C4, Kv− F ), and the value of the largest color class when it is as small as possible, ψ′ s(C4, Kv − F ), are both completely settled. This result follows from the solution to another interesting problem, namely that of finding (s, p)equitable edge-colorings of Kv. S. LI ET AL. /AUSTRALAS. J. COMBIN. 71 (1) (2018), 92–103 93