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 {\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

A one-to-one correspondence between potential solutions of the cluster deletion problem and the minimum sum coloring problem, and its application to P 4 -sparse graphs

In this note we show a one-to-one correspondence between potentially optimal solutions to the cluster deletion problem in a graph G and potentially optimal solutions for the minimum sum coloring problem in G (i.e. the complement graph of G). We apply this correspondence to polynomially solve the cluster deletion problem in a subclass of P 4 -sparse graphs that strictly includes P 4 -reducible graphs