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On the Circumference of 2-Connected $$mathcal{P}_{3}$$-Dominated Graphs

Authors:	Jiangyan Guo Elkin Vumar

Affiliation:	(1) College of Mathematic and System Sciences, Xinjiang University, Urumqi, 830046, P.R. China

Abstract:	Let G be a connected graph. For at distance 2, we define , and $J^{prime}(x, y) = {u\|u in N (x) cap N(y)$ , if then . G is quasi-claw-free if it satisfies , and G is P ₃-dominated( $mathcal{P}_{3}{mathcal{D}}$ ) if it satisfies $J(x,y)cup J^{prime} (x,y) neq emptyset$ , for every pair (x, y) of vertices at distance 2. Certainly ${mathcal{P}}_3 {mathcal{D}}$ contains as a subclass. In this paper, we prove that the circumference of a 2-connected P ₃-dominated graph G on n vertices is at least min or $G in {mathcal{F}} cup {K_{2,3}, K_{1,1,3}}$ , moreover if then G is hamiltonian or $G in {mathcal{F}}cup{K_{2,3}, K_{1,1,3}}$ , where is a class of 2-connected nonhamiltonian graphs.

Keywords:	Circumference Quasi-claw-free graph P ₃-Dominated graph Hamiltonian graph
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