On sufficient degree conditions for traceability of claw-free graphs

Abstract

We present new results on the traceability of claw-free graphs. In particular, we consider sufficient minimum degree and degree sum conditions that imply that these graphs admit a Hamilton path, unless they have a small order or they belong to well-defined classes of exceptional graphs. Our main result implies that a 2-connected claw-free graph G of sufficiently large order n with δ(G)≥3 is traceable if the degree sum of any set of t independent vertices of G is at least t(n+6)6, where t∈{1,2,…,6}, and that this lower bound n+66 on the degree sums is asymptotically sharp. Our results also imply that a 2-connected claw-free graph G of sufficiently large order n with minimum degree δ(G)≥22 is traceable if the degree sum of any set of t independent vertices of G is at least t(2n−5)14, where t∈{1,2,…,7}, unless G is a member of well-defined classes of exceptional graphs depending on t, and that this lower bound 2n−514 on the degree sums is asymptotically sharp. Our results also imply that a 2-connected claw-free graph G of sufficiently large order n with δ(G)≥18 is traceable if the degree sum of any set of 6 independent vertices is larger than n−6, and that this lower bound on the degree sums is sharp.