Face-Degree Bounds for Planar Critical Graphs

Ligang Jin,Yingli Kang,Eckhard Steffen

doi:10.37236/5895

Abstract

The only remaining case of a well known conjecture of Vizing states that there is no planar graph with maximum degree 6 and edge chromatic number 7. We introduce parameters for planar graphs, based on the degrees of the faces, and study the question whether there are upper bounds for these parameters for planar edge-chromatic critical graphs. Our results provide upper bounds on these parameters for smallest counterexamples to Vizing's conjecture, thus providing a partial characterization of such graphs, if they exist.For $k \leq 5$ the results give insights into the structure of planar edge-chromatic critical graphs.

Highlights

We consider finite simple graphs G with vertex set V (G) and edge set E(G)
Let G be a 2-connected planar graph, Σ be an embedding of G in the Euclidean plane and F (G) be the set of faces of (G, Σ)
We call bk the the average facedegree bound, and b∗k the local average face-degree bound for k-critical planar graphs

Summary

Introduction

We consider finite simple graphs G with vertex set V (G) and edge set E(G). The vertexdegree of v ∈ V (G) is denoted by dG(v), and ∆(G) denotes the maximum vertex-degree of G. Vizing’s conjecture has been proved for planar graph with maximum vertex-degree 7 by Grunewald [3], Sanders and Zhao [6], and Zhang [13] independently. Let G be a 2-connected planar graph, Σ be an embedding of G in the Euclidean plane and F (G) be the set of faces of (G, Σ). We call bk the the average facedegree bound, and b∗k the local average face-degree bound for k-critical planar graphs. Seymour’s exact conjecture [7] says that every critical planar graph is overfull If this conjecture is true for k ∈ {3, 4, 5}, bk is equal to the lower bound given in Theorem 1.

Preliminaries

Proofs

Theorem 2

Concluding remarks