A Linear Heterochromatic Number of Graphs

Abstract.Let G=(V(G),E(G)) be a multigraph with multiple loops allowed, and V0⊆V(G). We define h(G,V0) to be the minimum integer k such that for every edge-colouring of G using exactly k colours, all the edges incident with some vertex in V0 receive different colours. In this paper we prove that if each x∈V0 is incident to at least two edges of G, then h(G,V0)=φ(G[V0])+|E(G)|−|V0|+1 where φ(G[V0]) is the maximum cardinality of a set of mutually disjoint cycles (of length at least two) in the subgraph induced by V0.