Fully cycle extendability of -restricted graphs

A subgraph isomorphic to K 1 , p ( p ≥ 3 ) in a graph G is a p -claw of G . A graph G is K 1 , p -restricted if for any p -claw H of G the number of the edges in G among vertices of degree 1 in H is at least p − 2 . Clearly, every claw-free graph is K 1 , p -restricted. We prove that every connected, locally connected K 1 , 4 -restricted graph with minimum degree at least 3 is fully cycle extendable.