Edge coloring of graphs of signed class 1 and 2

Abstract

Recently, Behr [1] introduced a notion of the chromatic index of signed graphs and proved that for every signed graph (G, σ) it holds thatwhere ∆(G) is the maximum degree of G and χ ′ denotes its chromatic index.In general, the chromatic index of (G, σ) depends on both the underlying graph G and the signature σ. In the paper we study graphs G for which χ ′ (G, σ) does not depend on σ. To this aim we introduce two new classes of graphs, namely 1 ± and 2 ± , such that graph G is of class 1 ± (respectively, 2 ± ) if and only if χ ′ (G, σ) = ∆(G) (respectively, χ ′ (G, σ) = ∆(G) + 1) for all possible signatures σ.We prove that all wheels, necklaces, complete bipartite graphs K r,t with r = t and almost all cacti graphs are of class 1 ± . Moreover, we give sufficient and necessary conditions for a graph to be of class 2 ± , i.e. we show that these graphs must have odd maximum degree and give examples of such graphs with arbitrary odd maximum degree bigger that 1.