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

摘要: 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(P_n□T_m, σ)中，其中P_n和T_m分别是有n个顶点的路和有m个顶点的森林，当n > 2且∆(T_m) > 1时，则χ' (P_n□T_m, σ) = ∆(P_n□T_m, σ)。

Abstract: 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 (P_n□T_m, σ), Pn and T_m are respectively paths with the number of n vertices and forests with the number of m  vertices.  When  n > 2  and  ∆(T_m) > 1,  then  χ' (P_n□T_m, σ) = ∆(P_n□T_m, σ).