On the Chromatic Index of the Signed Generalized Petersen Graph GP(n,2)

Abstract

Let G be a graph and σ:E(G)→{+1,−1} be a mapping. The pair (G,σ), denoted by Gσ, is called a signed graph. A (proper) l-edge coloring γ of Gσ is a mapping from each vertex–edge incidence of Gσ to Mq such that γ(v,e)=−σ(e)γ(w,e) for each edge e=vw, and no two vertex–edge incidences have the same color; that is, γ(v,e)≠γ(v,f). The chromatic index is the minimal number q such that Gσ has a proper q-edge coloring, denoted by χ′(Gσ). In 2020, Behr proved that the chromatic index of a signed graph is its maximum degree or maximum plus one. In this paper, we considered the chromatic index of the signed generalized Petersen graph GP(n,2) and show that its chromatic index is its maximum degree for most cases. In detail, we proved that (1) χ′(GPσ(n,2))=3 if n≡3 mod 6(n≥9); (2) χ′(GPσ(n,2))=3 if n=2p(p≥4).