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arXiv · 0706.3848

Minimum Sum Edge Colorings of Multicycles

Jean Cardinal·Vlady Ravelomanana·Mario Valencia-Pabon
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Abstract

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.

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Jean Cardinal, Vlady Ravelomanana, Mario Valencia-Pabon. 2008-01-22. Minimum Sum Edge Colorings of Multicycles. https://arxiv.org/abs/0706.3848

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