Minimum Sum Edge Colorings of Multicycles

TL;DR

This paper investigates the minimum sum edge coloring problem for multicycles, providing a closed-form expression for the chromatic edge strength, and introduces algorithms for optimal coloring, extending to broader coloring problems.

Contribution

It derives a closed-form formula for the chromatic edge strength of multicycles and proposes algorithms for minimum sum edge coloring, extending Berge's theorem.

Findings

Chromatic edge strength of multicycles is explicitly characterized.

Minimum sum edge coloring can be achieved with the chromatic index.

Algorithms for optimal coloring of multicycles are provided.

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.