A generalization of Noel–Reed–Wu Theorem to signed graphs

Abstract

Let Σ be a signed graph where two edges joining the same pair of vertices with opposite signs are allowed. The zero-free chromatic number χ∗(Σ) of Σ is the minimum even integer 2k such that G admits a proper coloring f:V(Σ)↦{±1,±2,…,±k}. The zero-free list chromatic number χl∗(Σ) is the list version of zero-free chromatic number. Σ is called zero-free chromatic-choosable if χl∗(Σ)=χ∗(Σ). We show that if Σ has at most χ∗(Σ)+1 vertices then Σ is zero-free chromatic-choosable. This result strengthens Noel–Reed–Wu Theorem which states that every graph G with at most 2χ(G)+1 vertices is chromatic-choosable, where χ(G) is the chromatic number of G.