Hamilton Cycles in 10-Tough $(2P_2 \cup P_1)$-Free Graphs

2026-08-25

AI Summary

研究解决了10-tough $(2P_2 \cup P_1)$-free图中寻找哈密顿圈的问题，通过分析边的邻域大小，采用匹配路径覆盖压缩或非对称连通性分析方法。

Abstract

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.

Problem

• Hamilton Cycles
• 10-Tough
• 2P_2 ∪ P_1-Free Graphs

Innovation

• Hamiltonian cycle
• 10-tough graph
• matched path-cover
• asymmetric analysis
• cycle-extension lemma