Minimum sum edge colorings of multicycles

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 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.

• Type: Journal
• Year: 2010
• Where: DAM
• Authors: Jean Cardinal, Vlady Ravelomanana, Mario Valencia-Pabon
• Keywords: Chromatic Edge Strength, Coloring Problems, DAM 2010, Minimum Sum Edge