A conjecture on k - factor - critical and 3 - γ - critical 1 graphs ∗
نویسندگان
چکیده
4 For a graph G = (V,E), a set S ⊆ V is a dominating set if every vertex in 5 V is either in S or is adjacent to a vertex in S. The domination number γ(G) 6 of G is the minimum order of a dominating set in G. A graph G is said to be 7 domination vertex critical, if γ(G− v) < γ(G) for any vertex v in G. A graph 8 G is domination edge critical, if γ(G∪ e) < γ(G) for any edge e / ∈ E(G). We 9 call a graph G k-γ-vertex-critical (resp. k-γ-edge-critical) if it is domination 10 vertex critical (resp. domination edge critical) and γ(G) = k. Ananchuen and 11 Plummer posed the conjecture: Let G be a k-connected graph with the minimum 12 degree at least k+1, where k > 2 and k≡ |V | (mod 2). If G is 3-γ-edge-critical 13 and claw-free, then G is k-factor-critical. In this paper we present a proof to 14 this conjecture, and we also discuss the properties such as connectivity and 15 bicriticality in 3-γ-vertex-critical claw-free graph. 16
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