路和森林的符号乘积图的边染色

Edge Coloring of the Signed Product Graphs of Paths and Forests

2019 年, Behr 利用符号图的边染色概念证明了对于任意的符号图(G, σ)都有∆(G, σ) ≤ χ'(G, σ) ≤ ∆(G, σ) + 1, 其中χ'(G, σ)是(G, σ)的边染色数, ∆(G, σ) 是(G, σ)的最大度。 本文我们证明了在路和森林的符号乘积图(Pn□Tm, σ)中,其中Pn和Tm分别是有n个顶点的路和有m个顶点的森林,当n 】2且∆(Tm) 】1时,则χ'(Pn□Tm, σ) = ∆(Pn□Tm, σ)。

2019, Behr used the concept of edge coloring of signed graphs to prove that for any signed graphs (G, σ) there is ∆(G, σ) ≤ χ'(G, σ) ≤ ∆(G, σ)+1, where χ'(G, σ) is the number of edge coloring of (G, σ), ∆(G, σ) is the maximum degree of (G, σ). In this paper, we prove that in the signed product graphs of paths and forests (Pn□Tm, σ), Pn and Tm are respectively paths with the number of n vertices and forests with the number of m vertices. When n >2 and ∆(Tm) >1, then χ'(Pn□Tm, σ) = ∆(Pn□Tm, σ).