Strong Edge Coloring of Generalized Petersen Graphs

Ming Chen,Shan Zhou,Lianying Miao

doi:10.3390/math8081265

Ming Chen, Shan Zhou + Show 1 more

Open Access

https://doi.org/10.3390/math8081265

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Abstract

A strong edge coloring of a graph G is a proper edge coloring such that every color class is an induced matching. In 2018, Yang and Wu proposed a conjecture that every generalized Petersen graph P(n,k) with k≥4 and n>2k can be strong edge colored with (at most) seven colors. Although the generalized Petersen graph P(n,k) is a kind of special graph, the strong chromatic index of P(n,k) is still unknown. In this paper, we support the conjecture by showing that the strong chromatic index of every generalized Petersen graph P(n,k) with k≥4 and n>2k is at most 9.

Highlights

All graphs in this paper are finite and simple
Shiu, Wang and Chen [9] proved that every planar graph with a maximum degree of 4 can be strong edge colored with at most 19 colors
In 2018, Wu and Yang [15] conjectured that every generalized Petersen graph P(n, k) with n > 2k and k ≥ 4 can be strong edge colored with 7 colors

Summary

Introduction

All graphs in this paper are finite and simple. We denote the minimum and maximum degree of vertices in G by δ(G) and ∆(G), respectively. Shiu, Wang and Chen [9] proved that every planar graph with a maximum degree of 4 can be strong edge colored with at most 19 colors. In 1993, Brualdi and Massey [10] conjectured that every bipartite graph G(A, B) can be strong edge colored with at most ∆(A)∆(B) colors. In 2018, Wu and Yang [15] conjectured that every generalized Petersen graph P(n, k) with n > 2k and k ≥ 4 can be strong edge colored with 7 colors. They showed some upper bound of strong chromatic index of P(n, k) with k ∈ {1, 2, 3}. Every generalized Petersen graph P(n, k) with n > 2k and k ≥ 4 can be strong edge colored with at most 9 colors.

Lemmas

Coloring

Conclusions