坚韧图的闭包引理与哈密顿度条件
A Closure Lemma for tough graphs and Hamiltonian degree conditions
AI总结：
本文提出坚韧图的闭包引理，证明3t-1/2-坚韧图的哈密顿性等价于其t-闭包的哈密顿性，并据此证明Hoàng关于哈密顿度序列的猜想在t=4时成立。
摘要：
图 $G$ 的闭包是指从 $G$ 出发，反复在度之和至少为 $n$ 的不相邻顶点对之间加边所得到的图 $G^*$，其中 $n$ 是 $G$ 的顶点数。Bondy 和 Chvátal 证明的著名闭包引理指出：图 $G$ 是哈密顿图当且仅当其闭包 $G^*$ 是哈密顿图。该引理可用于证明哈密顿图论中的若干经典结果。我们证明了一个适用于坚韧图的闭包引理版本。若对图 $G$ 的任意顶点集 $S$，$G-S$ 的连通分量数至多为 $t |S|$，则称 $G$ 是 $t$-坚韧的。哈密顿图必然为 1-坚韧。反过来，Chvátal 猜想存在常数 $t$，使得每个 $t$-坚韧图都是哈密顿图。图 $G$ 的 $t$-闭包是指从 $G$ 出发，反复在度之和至少为 $n-t$ 的不相邻顶点对之间加边所得到的图 $G^{t*}$。我们证明，对 $t\geq 2$，一个 $\frac{3t-1}{2}$-坚韧图 $G$ 是哈密顿图当且仅当其 $t$-闭包 $G^{t*}$ 是哈密顿图。Hoàng 提出如下猜想：设 $G$ 是度序列为 $d_1 \leq d_2 \leq \ldots \leq d_n$ 的图；若 $G$ 是 $t$-坚韧的，且对所有 $i < \frac{n}{2}$，当 $d_i\leq i$ 时有 $d_{n-i+t}\geq n-i$，则 $G$ 是哈密顿图。该猜想类似于 Chvátal 关于哈密顿理想的著名定理。Hoàng 证明了 $t \leq 3$ 时的猜想。利用坚韧图的闭包引理，我们证明了 $t = 4$ 时的猜想。
英文摘要：
The closure of a graph $G$ is the graph $G^*$ obtained from $G$ by repeatedly adding edges between pairs of non-adjacent vertices whose degree sum is at least $n$, where $n$ is the number of vertices of $G$. The well-known Closure Lemma proved by Bondy and Chvátal states that a graph $G$ is Hamiltonian if and only if its closure $G^*$ is. This lemma can be used to prove several classical results in Hamiltonian graph theory. We prove a version of the Closure Lemma for tough graphs. A graph $G$ is $t$-tough if for any set $S$ of vertices of $G$, the number of components of $G-S$ is at most $t |S|$. A Hamiltonian graph must necessarily be 1-tough. Conversely, Chvátal conjectured that there exists a constant $t$ such that every $t$-tough graph is Hamiltonian. The {\it $t$-closure} of a graph $G$ is the graph $G^{t*}$ obtained from $G$ by repeatedly adding edges between pairs of non-adjacent vertices whose degree sum is at least $n-t$. We prove that, for $t\geq 2$, a $\frac{3t-1}{2}$-tough graph $G$ is Hamiltonian if and only if its $t$-closure $G^{t*}$ is. Hoàng conjectured the following: Let $G$ be a graph with degree sequence $d_1 \leq d_2 \leq \ldots \leq d_n$; then $G$ is Hamiltonian if $G$ is $t$-tough and, $\forall i <\frac{n}{2},\mbox{ if } d_i\leq i \mbox{ then } d_{n-i+t}\geq n-i$. This conjecture is analogous to the well known theorem of Chvátal on Hamiltonian ideals. Hoàng proved the conjecture for $t \leq 3$. Using the closure lemma for tough graphs, we prove the conjecture for $t = 4$.
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