Edge coloring of graphs of signed class 1 and 2

Abstract

Recently, Behr introduced a notion of the chromatic index of signed graphs and proved that for every signed graph $(G$, $σ)$ it holds that \[ Δ(G)\leqχ'(G\text{, }σ)\leqΔ(G)+1\text{,} \] where $Δ(G)$ 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 $1^\pm$ and $2^\pm$, such that graph $G$ is of class $1^\pm$ (respectively, $2^\pm$) if and only if $χ'(G$, $σ)=Δ(G)$ (respectively, $χ'(G$, $σ)=Δ(G)+1$) for all possible signatures $σ$. We prove that all wheels, necklaces, complete bipartite graphs $K_{r,t}$ with $r\neq t$ and almost all cacti graphs are of class $1^\pm$. Moreover, we give sufficient and necessary conditions for a graph to be of class $2^\pm$, 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 $1$.