Equipartite edge colouring of multigraphs

Abstract

Let G=(V, E) be a multigraph, without loops. For every vertex x, let E _x be the set of the edges of G that are incident to x. An edge colouring f of G is said to be an h-equipartite edge colouring of G, for a fixed h ⋯ N, h ≥ 2, if for every x⋯V such that \E_x\ = hq_x + r_x, 0 ≤ r_x < h, there exists a partition of E_x in q_x colour classes of cardinality h and one colour class of cardinality r_x. The maximum number fc for which there exists an h-equipartite edge k-colouring of G is denoted by X̄_h(G). In this paper we prove some results for 2-equipartite edge colourings. In particular we calculate X̄₂(G) when G is a complete graph or a complete bipartite graph. This paper can be considered as a continuation of [5].