具有指定最小度且无近生成圈的哈密顿图
Hamiltonian graphs with prescribed minimum degree and no near-spanning cycles
- East China Normal University（华东师范大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文构造性地解决了Häggkvist提出的问题，证明了对于给定最小度和足够大的阶数，存在不含近生成圈的哈密顿图，并提出了开放问题。
AI中文摘要：
1984年，Roland Häggkvist提出了一个关于构造阶数为n、最小度较大且不含(n-2)-圈的最小度至少为3的哈密顿图的问题。他提到自己不知道这样的图。我们通过证明以下两个结果来解决这个问题。(1) 对于每个整数d≥3和每个整数n≥15d-14，存在一个阶数为n、最小度为d且不含(n-2)-圈的哈密顿图。(2) 对于每个整数d≥3、每个正整数k以及每个整数n≥(k+1)[(d-1)(k+3)+1]，存在一个阶数为n、最小度为d且对任意s∈{1,2,…,k}都不含(n-s)-圈的哈密顿图。证明是构造性的。我们还提出了几个开放问题。
英文摘要：
In 1984, Roland Häggkvist posed the problem of constructing Hamiltonian graphs of order $n$ with large minimum degree and no $(n-2)$-cycle. He remarked that he did not know of such a graph with minimum degree at least three. We solve this problem by proving the following two results. (1) For every integer $d\ge 3$ and every integer $n\ge 15d-14,$ there exists a Hamiltonian graph of order $n$ and minimum degree $d$ that contains no $(n-2)$-cycle. (2) For every integer $d\ge 3,$ every positive integer $k,$ and every integer $n\ge (k+1)[(d-1)(k+3)+1],$ there exists a Hamiltonian graph of order $n$ and minimum degree $d$ that contains no $(n-s)$-cycle for any $s\in\{1,2,\dots,k\}.$ The proofs are constructive. We also pose several open problems.
