Edge coloring of graphs of signed class 1 and 2
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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)≤χ'(G, σ)≤Δ(G)+1, 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^± and 2^±, such that graph G is of class 1^± (respectively, 2^±) 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≠ 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.
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