Toughness and Hamiltonicity in Random Apollonian Networks

坚韧图中哈密顿性的度序列条件的一个加强

A strengthening of a degree sequence condition for Hamiltonicity in tough graphs

本文推广Chvátal的经典结果，验证了Hoàng提出的哈密顿性度序列猜想在所有t≥4的t-坚韧图中成立，证明基于此前建立的t≥4的坚韧度闭包引理，同时指出该引理在t=1时不成立。

Graph toughness from Laplacian eigenvalues

The toughness $t(G)$ of a graph $G=(V,E)$ is defined as $t(G)=\min\{\frac{|S|}{c(G-S)}\}$, in which the minimum is taken over all $S\subset V$ such that $G-S$ is disconnected, where $c(G-S)$ denotes the number of components of $G-S$. We present two tight lower bounds for $t(G)$ in terms of the Laplacian eigenvalues and provide strong support for a conjecture for a better bound which, if true, implies both bounds, and improves and generalizes known bounds by Alon, Brouwer, and the first author. As applications, several new results on perfect matchings, factors and walks from Laplacian eigenvalu…

The Toughness of Kneser Graphs

The \textit{toughness} $t(G)$ of a graph $G$ is a measure of its connectivity that is closely related to Hamiltonicity. Brouwer proved the lower bound $t(G) > \ell / λ- 2$ on the toughness of any connected $\ell$-regular graph, where $ λ$ is the largest nontrivial eigenvalue of the adjacency matrix. He conjectured that this lower bound can be improved to $\ell / λ-1$ and this conjecture is still open. Brouwer also observed that many families of graphs (in particular, those achieving equality in the Hoffman ratio bound for the independence number) have toughness exactly $\ell / λ$. Cioabă and W…

稀疏化与喷洒：从鲁棒采样到几乎哈密顿性

Thinning and sprinkling: from robust sampling to almost Hamiltonicity

本文提出稀疏化-喷洒技术，证明坚韧图等图类的随机子图保持鲁棒性，并据此构造几乎哈密顿圈，解决Chvátal和Lovász的长期猜想。

Disconnected Forbidden Subgraphs, Toughness and Hamilton Cycles

In 1974, Goodman and Hedetniemi proved that every 2-connected $(K_{1,3},K_{1,3}+e)$-free graph is hamiltonian. This result gave rise many other hamiltonicity conditions for various pairs and triples of forbidden connected subgraphs under additional connectivity conditions. In 1997, it was proved that a single forbidden connected subgraph $R$ in 2-connected graphs can create only a trivial class of hamiltonian graphs (complete graphs) with $R=P_3$. In this paper we prove that a single forbidden subgraph $R$ can create a non trivial class of hamiltonian graphs if $R$ is disconnected: $(\ast1)$ e…

The relation between Hamiltonian and $1$-tough properties of the Cartesian product graphs

The relation between Hamiltonicity and toughness of a graph is a long standing research problem. The paper studies the Hamiltonicity of the Cartesian product graph $G_1\square G_2$ of graphs $G_1$ and $G_2$ satisfying that $G_1$ is traceable and $G_2$ is connected with a path factor. Let Pn be the path of order $n$ and $H$ be a connected bipartite graph. With certain requirements of $n$, we show that the following three statements are equivalent: (i) $P_n\square H$ is Hamiltonian; (ii) $P_n\square H$ is $1$-tough; and (iii) $H$ has a path factor.

Hamiltonicity of $1$-tough $(P_2\cup kP_1)$-free graphs

Given a graph $H$, a graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. For a positive real number $t$, a non-complete graph $G$ is said to be $t$-tough if for every vertex cut $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. A complete graph is said to be $t$-tough for any $t>0$. Chvátal's toughness conjecture, stating that there exists a constant $t_0$ such that every $t_0$-tough graph with at least three vertices is Hamiltonian, is still open in general. Chvátal and Erdös \cite{CE} proved that, for any integer $k\ge 1$, every $\max\…