A Study of Sufficient Conditions for Hamiltonian Cycles

A graph G is Hamiltonian if it has a spanning cycle. The problem of determining if a graph is Hamiltonian is well known to be NP-complete. While there are several necessary conditions for Hamiltonicity, the search continues for sufficient conditions. In their paper, On Smallest Non-Hamiltonian Regular Tough Graphs (Congressus Numerantium 70), Bauer, Broersma, and Veldman stated, without a formal proof, that all 4-regular, 2-connected, 1-tough graphs on fewer than 18 nodes are Hamiltonian. They also demonstrated that this result is best possible. Following a brief survey of some sufficient conditions for Hamiltonicity, Bauer, Broersma, and Veldman\u27s result is demonstrated to be true for graphs on fewer than 16 nodes. Possible approaches for the proof of the n=16 and n=17 cases also will be discussed

On some intriguing problems in Hamiltonian graph theory -- A survey

We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, $t$-tough graphs, and claw-free graphs

Dirac's theorem for random regular graphs

We prove a `resilience' version of Dirac's theorem in the setting of random regular graphs. More precisely, we show that, whenever $d$ is sufficiently large compared to $\varepsilon>0$, a.a.s. the following holds: let $G'$ be any subgraph of the random $n$-vertex $d$-regular graph $G_{n,d}$ with minimum degree at least $(1/2+\varepsilon)d$. Then $G'$ is Hamiltonian. This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly, the condition that $d$ is large cannot be omitted, and secondly, the minimum degree bound cannot be improved.Comment: Final accepted version, to appear in Combinatorics, Probability & Computin

Some local--global phenomena in locally finite graphs

In this paper we present some results for a connected infinite graph $G$ with finite degrees where the properties of balls of small radii guarantee the existence of some Hamiltonian and connectivity properties of $G$. (For a vertex $w$ of a graph $G$ the ball of radius $r$ centered at $w$ is the subgraph of $G$ induced by the set $M_r(w)$ of vertices whose distance from $w$ does not exceed $r$). In particular, we prove that if every ball of radius 2 in $G$ is 2-connected and $G$ satisfies the condition $d_G(u)+d_G(v)\geq |M_2(w)|-1$ for each path $uwv$ in $G$, where $u$ and $v$ are non-adjacent vertices, then $G$ has a Hamiltonian curve, introduced by K\"undgen, Li and Thomassen (2017). Furthermore, we prove that if every ball of radius 1 in $G$ satisfies Ore's condition (1960) then all balls of any radius in $G$ are Hamiltonian.Comment: 18 pages, 6 figures; journal accepted versio

Local resilience and Hamiltonicity Maker-Breaker games in random-regular graphs

For an increasing monotone graph property \mP the \emph{local resilience} of a graph $G$ with respect to \mP is the minimal $r$ for which there exists of a subgraph $H\subseteq G$ with all degrees at most $r$ such that the removal of the edges of $H$ from $G$ creates a graph that does not possesses \mP. This notion, which was implicitly studied for some ad-hoc properties, was recently treated in a more systematic way in a paper by Sudakov and Vu. Most research conducted with respect to this distance notion focused on the Binomial random graph model \GNP and some families of pseudo-random graphs with respect to several graph properties such as containing a perfect matching and being Hamiltonian, to name a few. In this paper we continue to explore the local resilience notion, but turn our attention to random and pseudo-random \emph{regular} graphs of constant degree. We investigate the local resilience of the typical random $d$-regular graph with respect to edge and vertex connectivity, containing a perfect matching, and being Hamiltonian. In particular we prove that for every positive $\epsilon$ and large enough values of $d$ with high probability the local resilience of the random $d$-regular graph, \GND, with respect to being Hamiltonian is at least $(1-\epsilon)d/6$. We also prove that for the Binomial random graph model \GNP, for every positive $\epsilon>0$ and large enough values of $K$, if $p>\frac{K\ln n}{n}$ then with high probability the local resilience of \GNP with respect to being Hamiltonian is at least $(1-\epsilon)np/6$. Finally, we apply similar techniques to Positional Games and prove that if $d$ is large enough then with high probability a typical random $d$-regular graph $G$ is such that in the unbiased Maker-Breaker game played on the edges of $G$, Maker has a winning strategy to create a Hamilton cycle.Comment: 34 pages. 1 figur