Equipartite edge colouring of multigraphs

Let G=(V,E) be a multigraph, without loops. For every vertex x, let Ex 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-eguipartite edge colouring of G, for a fixed h is an element of N, h >= 2, if for every x is an element of V such that vertical bar E(x vertical bar) = 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 k for which there exists an h-equipartite edge k-colouring of G is denoted (chi)over bar(h) (G). In this paper we prove some results for 2-equipartite edge colourings. In particular we calculate (chi)over bar(2) (G) when G is a complete graph or a complete bipartite graph. This paper can be considered as a continuation of [5].