Longest Paths and Cycles in Connected Claw-Free Graphs

【Abstract】 A graph is called claw-free if it does not contain a claw as its induced subgraph.In this paper, we prove the following results:1)If G is a 2-connected claw-free graph on n vertices,then for any vertex v and any two distinct vertices x and y in V(G)-{v},G has a path containing v and all neighbors of v and connecting x and y;2) Let C be the longest cycle in a 3-connected claw-free graph G and H a component of G-C,and if H is connected but not 2-connected,then there exist nonadjacent vertices u and v in H such that |V(C)|≥(3(d(u)+)d(v))-2.更多还原