Minimum Sum Edge Colorings of Multicycles

Jean Cardinal (1) , Vlady Ravelomanana (2) , Mario Valencia-Pabon (2)

1 ULB - Université libre de Bruxelles
2 LIPN - Laboratoire d'Informatique de Paris-Nord

In the minimum sum edge coloring problem, we aim to assign natural numbers to edges of a graph, so that adjacent edges receive different numbers, and the sum of the numbers assigned to the edges is minimum. The {\em chromatic edge strength} of a graph is the minimum number of colors required in a minimum sum edge coloring of this graph. We study the case of multicycles, defined as cycles with parallel edges, and give a closed-form expression for the chromatic edge strength of a multicycle, thereby extending a theorem due to Berge. It is shown that the minimum sum can be achieved with a number of colors equal to the chromatic index. We also propose simple algorithms for finding a minimum sum edge coloring of a multicycle. Finally, these results are generalized to a large family of minimum cost coloring problems.

Mots clés: graph coloring; minimum sum coloring; chromatic strength

Domaines: Mathématique discrète [cs.DM]

Jean Cardinal, Vlady Ravelomanana, Mario Valencia-Pabon. Minimum Sum Edge Colorings of Multicycles. 2008. ⟨hal-00155565v2⟩