Minimum sum edge colorings of multicycles

作者：Cardinal Jean^*; Ravelomanana Vlady; Valencia Pabon Mario

来源：Discrete Applied Mathematics, 2010, 158(12): 1216-1223.

DOI：10.1016/j.dam.2009.04.020

• Graph coloring
• Minimum sum coloring
• Chromatic strength

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.

• 出版日期2010-6-28