Edge coloring of graphs of signed class 1 and 2

Recently, Behr (2020) introduced a notion of the chromatic index of signed graphs and proved that for every signed graph (G, & sigma;) it holds that increment (G) & LE; & chi;& PRIME;(G,& sigma;) & LE; increment (G)+ 1,where increment (G) is the maximum degree of G and & chi;& PRIME; denotes its chromatic index. In general, the chromatic index of (G, & sigma;) depends on both the underlying graph G and the signature & sigma;. In the paper we study graphs G for which & chi;& PRIME;(G, & sigma;) does not depend on & sigma;. To this aim we introduce two new classes of graphs, namely 1 & PLUSMN; and 2 & PLUSMN;, such that graph G is of class 1 & PLUSMN; (respectively, 2 & PLUSMN;) if and only if & chi;& PRIME;(G, & sigma;)= increment (G) (respectively, & chi;& PRIME;(G, & sigma;)= increment (G)+ 1) for all possible signatures & sigma;. We prove that all wheels, necklaces, complete bipartite graphs Kr,t with r= t and almost all cacti graphs are of class 1 & PLUSMN;. Moreover, we give sufficient and necessary conditions for a graph to be of class 2 & PLUSMN;, i.e. we show that these graphs must have odd maximum degree and give examples of such graphs with arbitrary odd maximum degree bigger than 1.& COPY; 2023 Elsevier B.V. All rights reserved.