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 + 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.

中文翻译：

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无爪图溯源的充分度条件

摘要 我们提出了关于无爪图可追溯性的新结果。特别是，我们考虑了足够的最小度和度和条件，这意味着这些图允许汉密尔顿路径，除非它们的阶数很小或属于定义明确的异常图类。我们的主要结果表明，如果 G 的任何 t 个独立顶点集的度和至少为 t ( n + 6 )，则具有 δ ( G ) ≥ 3 的足够大阶 n 的 2 连通无爪图 G 是可追踪的6 ，其中 t ∈ { 1 , 2 , … , 6 } ，并且该度数和的下界 n + 6 6 渐近尖锐。我们的结果还意味着，如果 G 的任何 t 个独立顶点集的度数和至少为 t ( 2 n )，则具有最小度数 δ ( G ) ≥ 22 的足够大阶 n 的 2 连通无爪图 G 是可追踪的− 5 ) 14 , 其中 t ∈ { 1 , 2 , … , 7 } ，除非 G 是依赖于 t 的定义明确的异常图类的成员，并且度和的这个下界 2 n − 5 14 是渐近尖锐的。我们的结果还意味着，如果任何 6 个独立顶点集的度数总和大于 n − 6 ，则具有足够大阶 n 且 δ ( G ) ≥ 18 的 2 连通无爪图 G 是可追踪的，并且这个较低度数和的界限是尖锐的。