Related papers: Loose Hamilton Cycles in Random Uniform Hypergraph…

It was shown by Kutnar and \v Sparl in 2009 that every connected vertex-transitive graph of order $6p$, where $p$ is a prime, contains a Hamilton path. In this paper, it will be shown that every such graph contains a Hamilton cycle, except…

We consider a game played on an initially empty graph where two players alternate drawing an edge between vertices subject to the condition that no degree can exceed $k$. We show that for $k=3$, either player can avoid a Hamilton cycle, and…

Let $\mu > 2$ and $\epsilon > 0$. We show that, if $G$ is a sufficiently large simple graph of average degree at least $\mu$, and $H$ is a random spanning subgraph of $G$ formed by including each edge independently with probability $p \ge…

In 1970 Lov\'asz conjectured that every connected vertex-transitive graph admits a Hamilton cycle, apart from five exceptional graphs. This conjecture has recently been settled for graphs defined by intersecting set systems, which feature…

A fullerene graph is a 3-connected cubic planar graph with pentagonal and hexagonal faces. The leapfrog transformation of a planar graph produces the trucation of the dual of the given graph. A fullerene graph is leapfrog if it can be…

A graph $G$ is $l$-path Hamiltonian if every path of length not exceeding $l$ is contained in a Hamiltonian cycle. It is well known that a 2-connected, $k$-regular graph $G$ on at most $3k-1$ vertices is edge-Hamiltonian if for every edge…

A step forward is made in a long standing Lov\'{a}sz's problem regarding hamiltonicity of vertex-transitive graphs by showing that every connected vertex-transitive graph of order a product of two primes, other than the Petersen graph,…

In the standard random graph process, edges are added to an initially empty graph one by one uniformly at random. A classic result by Ajtai, Koml\'os, and Szemer\'edi, and independently by Bollob\'as, states that in the standard random…

We study the existence of directed Hamilton cycles in random digraphs with $m$ edges where we condition on minimum in- and out-degree $\d \ge k+1$, where $k \ge 1$. Denote such a random graph by $D_{n,m}^{(\delta\geq k+1)}$. Let $m=cn$ and…

A graph on $n$ vertices is called pancyclic if it contains a cycle of every length $3\le l \le n$. Given a Hamiltonian graph $G$ with independence number at most $k$ we are looking for the minimum number of vertices $f(k)$ that guarantees…

We show that provided $\log^{50} n/n \leq p \leq 1 - n^{-1/4}\log^9 n$ we can with high probability find a collection of $\lfloor \delta(G)/2 \rfloor$ edge-disjoint Hamilton cycles in $G \sim G_{n, p}$, plus an additional edge-disjoint…

Let $G_{k,n}$ be the $n$-balanced $k$-partite graph, whose vertex set can be partitioned into $k$ parts, each has $n$ vertices. In this paper, we prove that if $k \geq 2,n \geq 1$, for the edge set $E(G)$ of $G_{k,n}$ $$|E(G)|…

Let $H$ be an edge colored hypergraph. We say that $H$ contains a \emph{rainbow} copy of a hypergraph $S$ if it contains an isomorphic copy of $S$ with all edges of distinct colors. We consider the following setting. A randomly edge colored…

As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion…

In this paper, we consider a random geometric graph (RGG)~\(G\) on~\(n\) nodes with adjacency distance~\(r_n\) just below the Hamiltonicity threshold and construct Hamiltonian cycles using additional edges called bridges. The bridges by…

For all integers $k$ with $k\geq 2$, if $G$ is a balanced $k$-partite graph on $n\geq 3$ vertices with minimum degree at least \[…

The generalized Petersen graph $G(n, k)$ is a cubic graph with vertex set $V(G(n, k)) = \{v_i\}_{0 \leq i < n} \cup \{w_i\}_{0 \leq i < n}$ and edge set $E(G(n, k)) = \{v_i v_{i+1}\}_{0 \leq i < n} \cup \{w_i w_{i+k}\}_{0 \leq i < n} \cup…

Given a graph on $n$ vertices and an assignment of colours to the edges, a rainbow Hamilton cycle is a cycle of length $n$ visiting each vertex once and with pairwise different colours on the edges. Similarly (for even $n$) a rainbow…

Let G_{\rm 3-out} denote the random graph on vertex set [n] in which each vertex chooses 3 neighbors uniformly at random. Note that G_{\rm 3-out} has minimum degree 3 and average degree 6. We prove that the probability that G_{\rm 3-out} is…

A $3$-uniform loose cycle, denoted by $C_t$, is a $3$-graph on $t$ vertices whose vertices can be arranged cyclically so that each hyperedge consists of three consecutive vertices, and any two consecutive hyperedges share exactly one…