Finding Hamiltonian cycles in [formula omitted]-free graphs with bounded Dilworth numbers

Abstract

Let u and v be two vertices in a graph G . We say vertex u dominates vertex v if N ( v ) ⊆ N ( u ) ∪ { u } . If u dominates v or v dominates u , then u and v are comparable. The Dilworth number of a graph G , denoted as D i l ( G ) , is the largest number of pairwise incomparable vertices in the graph G . A graph G is called quasi-claw-free if it satisfies the property: d ( x , y ) = 2 ⇒ there exists u ∈ N ( x ) ∩ N ( y ) such that N [ u ] ⊆ N [ x ] ∪ N [ y ] . A graph is called { q u a s i - c l a w , K 1 , 5 , K 1 , 5 + e } -free if it is quasi-claw-free and contains no induced subgraph isomorphic to K 1 , 5 or K 1 , 5 + e , where K 1 , 5 + e is a graph obtained by joining a pair of nonadjacent vertices in K 1 , 5 . It is shown that if G is a k ( k ≥ 2 )-connected { q u a s i - c l a w , K 1 , 5 , K 1 , 5 + e } -free graph with D i l ( G ) ≤ 2 k − 1 , then G is Hamiltonian and a Hamiltonian cycle in G can be found in polynomial time.