Independence number of edge‐chromatic critical graphs

Abstract

Let G $G$ be a simple graph with maximum degree Δ ( G ) ${\rm{\Delta }}(G)$ and chromatic index χ ′ ( G ) $\chi ^{\prime} (G)$ . A classical result of Vizing shows that either χ ′ ( G ) = Δ ( G ) $\chi ^{\prime} (G)={\rm{\Delta }}(G)$ or χ ′ ( G ) = Δ ( G ) + 1 $\chi ^{\prime} (G)={\rm{\Delta }}(G)+1$ . A simple graph G $G$ is called edge- Δ ${\rm{\Delta }}$ -critical if G $G$ is connected, χ ′ ( G ) = Δ ( G ) + 1 $\chi ^{\prime} (G)={\rm{\Delta }}(G)+1$ and χ ′ ( G − e ) = Δ ( G ) $\chi ^{\prime} (G-e)={\rm{\Delta }}(G)$ for every e ∈ E ( G ) $e\in E(G)$ . Let G $G$ be an n $n$ -vertex edge- Δ ${\rm{\Delta }}$ -critical graph. Vizing conjectured that α ( G ) $\alpha (G)$ , the independence number of G $G$ , is at most n 2 $\frac{n}{2}$ . The current best result on this conjecture, shown by Woodall, is α ( G ) < 3 n 5 $\alpha (G)\lt \frac{3n}{5}$ . We show that for any given ε ∈ ( 0 , 1 ) $\varepsilon \in (0,1)$ , there exist positive constants d 0 ( ε ) ${d}_{0}(\varepsilon )$ and D 0 ( ε ) ${D}_{0}(\varepsilon )$ such that if G $G$ is an n $n$ -vertex edge- Δ ${\rm{\Delta }}$ -critical graph with minimum degree at least d 0 ${d}_{0}$ and maximum degree at least D 0 ${D}_{0}$ , then α ( G ) < 1 2 + ε n $\alpha (G)\lt \left(\frac{1}{2}+\varepsilon \right)n$ . In particular, we show that if G $G$ is an n $n$ -vertex edge- Δ ${\rm{\Delta }}$ -critical graph with minimum degree at least d $d$ and Δ ( G ) ≥ ( d + 1 ) 4.5 d + 11.5 ${\rm{\Delta }}(G)\ge {(d+1)}^{4.5d+11.5}$ , then α ( G ) < 7 n 12 if d = 3 , 4 n 7 if d = 4 , d + 2 + ( d − 1 ) d 3 2 d + 4 + ( d − 1 ) d 3 n < 4 n 7 if d ≥ 19 . $\alpha (G)\lt \left.\left\{\displaystyle \begin{array}{cc}\frac{7n}{12} & \,\text{if}\,\,d=3,\\ \frac{4n}{7} & \,\text{if}\,\,d=4,\\ \frac{d+2+\sqrt[3]{(d-1)d}}{2d+4+\sqrt[3]{(d-1)d}}n\lt \frac{4n}{7} & \,\text{if}\,\,d\ge 19.\end{array}\right.$