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

Abstract In this paper we consider the existence of ( s , p )-equitable block-colorings of 4-cycle decompositions of K v − F , where F is a 1-factor of K v . 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 χ p ′ ( v ) and χ p ′ ( v ) , namely the least and greatest values of s for which there exists such a block-coloring of some 4-cycle decomposition of K v − 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 χ p ′ ( v ) when p ∈ { 2 , 3 , 4 } and χ 2 ′ ( v ) , and to provide good upper bounds on χ 3 ′ ( v ) .