Neighborhood intersections and Hamiltonicity in almost claw-free graphs

Abstract

Let G be a graph. The partially square graph G ∗ of G is a graph obtained from G by adding edges uv satisfying the conditions uv∉ E( G), and there is some w∈ N( u)∩ N( v), such that N( w)⊆ N( u)∪ N( v)∪{ u, v}. Let t>1 be an integer and Y⊆ V( G), denote n(Y)=|{v∈V(G) | min y∈Y{ dist G(v,y)}⩽2}|, I t(G)={Z | Z is an independent set of G,| Z|= t}. In this paper, we show that a k-connected almost claw-free graph with k⩾2 is hamiltonian if ∑ z∈ Z d( z)⩾ n( Z)− k in G for each Z∈I k+1(G ∗) , thereby solving a conjecture proposed by Broersma, Ryjác̆ek and Schiermeyer. Zhang's result is also generalized by the new result.