A Fan-type condition for cycles in 1-tough and k-connected (P2 ∪ kP1)-free graphs

IF 3.5 2区 数学 Q1 MATHEMATICS, APPLIED Applied Mathematics and Computation Pub Date : 2025-01-17 DOI:10.1016/j.amc.2025.129300
Zhiquan Hu, Jie Wang, Changlong Shen
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Abstract

For a graph G, define μk(G):=min{maxxSdG(x):SSk}, where Sk is the set consisting of all independent sets {u1,,uk} of G such that some vertex, say ui (1ik), is at distance two from every other vertex in it. A graph G is called 1-tough if for each cut set SV(G), GS has no more than |S| components. Recently, Shi and Shan [19] conjectured that for each integer k4, being 2k-connected is sufficient for 1-tough (P2kP1)-free graphs to be hamiltonian, which was confirmed by Xu et al. [20] and Ota and Sanka [16], respectively. In this article, we generalize the above results through the following Fan-type theorem: If G is a 1-tough and k-connected (P2kP1)-free graph and satisfies μk+1(G)7k65, where k2 is an integer, then G is hamiltonian or the Petersen graph.
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1-坚韧k-连通(P2 ∪ kP1)自由图中圈的一个fan型条件
对于图 G,定义 μk(G):=min{maxx∈SdG(x):S∈Sk},其中 Sk 是由 G 的所有独立集 {u1,...,uk}组成的集合,使得某个顶点,例如 ui (1≤i≤k),与其中的每个其他顶点的距离都是 2。如果对于每个切集 S⊆V(G),G-S 的分量不超过 |S|,则图 G 称为 1-韧图。最近,Shi 和 Shan [19]猜想,对于每个整数 k≥4,2k-连通足以使无 1-韧(P2∪kP1)图成为哈密顿图,这一点分别被 Xu 等人 [20] 和 Ota 和 Sanka [16] 所证实。在本文中,我们通过下面的范型定理来推广上述结果:如果 G 是一个 1韧且 k 连接的 (P2∪kP1)-free 图,并且满足 μk+1(G)≥7k-65,其中 k≥2 是整数,那么 G 是哈密顿图或彼得森图。
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来源期刊
CiteScore
7.90
自引率
10.00%
发文量
755
审稿时长
36 days
期刊介绍: Applied Mathematics and Computation addresses work at the interface between applied mathematics, numerical computation, and applications of systems – oriented ideas to the physical, biological, social, and behavioral sciences, and emphasizes papers of a computational nature focusing on new algorithms, their analysis and numerical results. In addition to presenting research papers, Applied Mathematics and Computation publishes review articles and single–topics issues.
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