Maximum average degree of list-edge-critical graphs and Vizing's conjecture

Abstract

Vizing conjectured that χ ′ ℓ ( G )≤ Δ + 1 for all graphs. For a graph G and nonnegative integer k , we say G is a k -list-edge-critical graph if χ ′ ℓ ( G )> k , but χ ′ ℓ ( G − e )≤ k for all e ∈ E ( G ) . We use known results for list-edge-critical graphs to verify Vizing’s conjecture for G with m a d ( G )<( Δ + 3)/2 and Δ ≤ 9 .