Abstract.
Let G=(V(G),E(G)) be a multigraph with multiple loops allowed, and V 0⊆V(G). We define h(G,V 0) 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 V 0 receive different colours. In this paper we prove that if each x∈V 0 is incident to at least two edges of G, then h(G,V 0)=φ(G[V 0])+|E(G)|−|V 0|+1 where φ(G[V 0]) is the maximum cardinality of a set of mutually disjoint cycles (of length at least two) in the subgraph induced by V 0.
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Acknowledgments. We thank the referee for suggesting us a short alternative proof of our main theorem.
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Montellano-Ballesteros, J., Neumann-Lara, V. A Linear Heterochromatic Number of Graphs. Graphs and Combinatorics 19, 533–536 (2003). https://doi.org/10.1007/s00373-003-0511-6
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DOI: https://doi.org/10.1007/s00373-003-0511-6