10-坚韧度的$(2P_2 \bigcup P_1)$-无图中的哈密顿回路
Hamilton Cycles in 10-Tough $(2P_2 \cup P_1)$-Free Graphs
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中文总结 AI 辅助
本文证明，每个至少含3个顶点的有限简单10-坚韧度$(2P_2 \bigcup P_1)$-无图都存在哈密顿回路，通过按边的联合邻域大小拆分情形分析完成证明。
中文摘要 AI 辅助
若一个图满足两个条件，则称其为10-坚韧度的$(2P_2 \bigcup P_1)$-无图：一是删除任意顶点集后，剩余分量数至少为2时，该顶点集的基数至少为剩余分量数的10倍；二是该图不含由两条不相交边和一个孤立顶点构成的诱导子图。本文证明，每个至少含3个顶点的有限简单10-坚韧度$(2P_2 \bigcup P_1)$-无图都是哈密顿图。证明过程按某条边的联合邻域阶是否不超过$4n/11$拆分：在小邻域情形下，将匹配路径覆盖压缩为指定匹配；在大邻域情形下，对假设的小割集留下的两个分量进行非对称分析，得到所需的连通性界。随后，通过指定边的哈密顿回路被扩展，再利用回路扩展引理插入剩余顶点。
英文摘要
A graph is called 10-tough and $(2P_2 \cup P_1)$-free if every vertex set whose deletion leaves at least two components has cardinality at least ten times the number of those components and if the graph has no induced subgraph consisting of two disjoint edges and an isolated vertex. We prove that every finite simple 10-tough $(2P_2 \cup P_1)$-free graph on at least three vertices is Hamiltonian. The proof splits according to whether some edge has joint neighbourhood of order at most $4n/11$. In the small-neighbourhood case, a matched path-cover is compressed to a prescribed matching. In the large-neighbourhood case, an asymmetric analysis of the two components left by a putative small cut yields the required connectivity bound. A Hamilton cycle through the prescribed edges is then expanded, and a cycle-extension lemma inserts the remaining vertices.
