Chvátal–Erdös Type Conditions for Hamiltonicity of Claw-Free Graphs

Abstract

For a graph H, let $$\alpha (H)$$ź(H) and $$\alpha ^{\prime }(H)$$źź(H) denote the independence number and the matching number, respectively. Let $$k\ge 2$$kź2 and $$r>0$$r>0 be given integers. We prove that if H is a k-connected claw-free graph with $$\alpha (H)\le r$$ź(H)≤r, then either H is Hamiltonian or the Ryja cź ek's closure $$cl(H)=L(G)$$cl(H)=L(G) where G can be contracted to a k-edge-connected $$K_3$$K3-free graph $$G_0^{\prime }$$G0ź with $$\alpha ^{\prime }(G_0^{\prime })\le r$$źź(G0ź)≤r and $$|V(G_0^{\prime })|\le \max \{3r-5, 2r+1\}$$|V(G0ź)|≤max{3r-5,2r+1} if $$k\ge 3$$kź3 or $$|V(G_0^{\prime })|\le \max \{4r-5, 2r+1\}$$|V(G0ź)|≤max{4r-5,2r+1} if $$k=2$$k=2 and $$G_0^{\prime }$$G0ź does not have a dominating closed trail containing all the vertices that are obtained by contracting nontrivial subgraphs. As corollaries, we prove the following:(a)A 2-connected claw-free graph H with $$\alpha (H)\le 3$$ź(H)≤3 is either Hamiltonian or $$cl(H)=L(G)$$cl(H)=L(G) where G is obtained from $$K_{2,3}$$K2,3 by adding at least one pendant edge on each degree 2 vertex;(b)A 3-connected claw-free graph H with $$\alpha (H)\le 7$$ź(H)≤7 is either Hamiltonian or $$cl(H)=L(G)$$cl(H)=L(G) where G is a graph with $$\alpha ^{\prime }(G)=7$$źź(G)=7 that is obtained from the Petersen graph P by adding some pendant edges or subdividing some edges of P. Case (a) was first proved by Xu et al. [19]. Case (b) is an improvement of a result proved by Flandrin and Li [12]. For a given integer $$r>0$$r>0, the number of graphs of order at most $$\max \{4r-5, 2r+1\}$$max{4r-5,2r+1} is fixed. The main result implies that improvements to case (a) or (b) by increasing the value of r and by enlarging the collection of exceptional graphs can be obtained with the help of a computer. Similar results involved degree or neighborhood conditions are also discussed.