On sufficient degree conditions for traceability of claw-free graphs

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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 + 6 6 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 ( 2 n − 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 2 n − 5 14 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.