A conjecture on k-factor-critical and 3-γ-critical graphs

Abstract

For a graph G = (V,E), a subset S ⊆ V is a dominating set if every vertex in V is either in S or is adjacent to a vertex in S. The domination number γ(G) of G is the minimum order of a dominating set in G. A graph G is said to be domination vertex critical, if γ(G − v) < γ(G) for any vertex v in G. A graph G is domination edge critical, if γ(G ∪ e) < γ(G) for any edge e ∉ E(G). We call a graph G k-γ-vertex-critical (resp. k-γ-edge-critical) if it is domination vertex critical (resp. domination edge critical) and γ(G) = k. Ananchuen and Plummer posed the conjecture: Let G be a k-connected graph with the minimum degree at least k+1, where k ⩾ 2 and k ≡ |V| (mod 2). If G is 3-γ-edge-critical and claw-free, then G is k-factor-critical. In this paper we present a proof to this conjecture, and we also discuss the properties such as connectivity and bicriticality in 3-γ-vertex-critical claw-free graph.