Edge Coloring of the Signed Generalized Petersen Graph

Cai, Hongyan; Sun, Qiang; Xu, Guangjun; Zheng, Shanshan

doi:10.1007/s40840-021-01212-w

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2022

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“…Yang et al [17] studied the strong chromatic index of GP(n, k) when 1 ≤ k ≤ 3. In [18], Cai et al studied the edge coloring of the generalized Petersen graph and got the following results. When n ≥ 5, the chromatic index of GP(n, 1) is three.…”

Section: Introductionmentioning

confidence: 99%

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

Zheng

Cai

Wang

et al. 2022

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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).

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Section: Introductionmentioning

confidence: 99%