Related papers: Some cyclic properties of $L_1$-graphs

An st-path is a path with the end-vertices s and t. An s-path is a path with an end-vertex s. The results of this paper include necessary and sufficient conditions for a {claw, net}-free graph G with given two different vertices s, t and an…

We study Hamiltonicity in graphs obtained as the union of a deterministic $n$-vertex graph $H$ with linear degrees and a $d$-dimensional random geometric graph $G^d(n,r)$, for any $d\geq1$. We obtain an asymptotically optimal bound on the…

A graph is $1$-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges…

A conjecture of Jackson from 1981 states that every $d$-regular oriented graph on $n$ vertices with $n\leq 4d+1$ is Hamiltonian. We prove this conjecture for sufficiently large $n$. In fact we prove a more general result that for all…

Let $D$ be a strongly connected directed graph of order $n\geq 4$ which satisfies the following condition (*): for every pair of non-adjacent vertices $x, y$ with a common in-neighbour $d(x)+d(y)\geq 2n-1$ and $min \{ d(x), d(y)\}\geq n-1$.…

We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lov\'{a}sz from 1969 and Thomassen from 1978, respectively, states that all connected vertex-transitive graphs…

We show that for each \eta>0 every digraph G of sufficiently large order n is Hamiltonian if its out- and indegree sequences d^+_1\le ... \le d^+_n and d^- _1 \le ... \le d^-_n satisfy (i) d^+_i \geq i+ \eta n or d^-_{n-i- \eta n} \geq n-i…

In 1973, Chv\'atal conjectured that there exists a constant $t_0$ such that every $t_0$-tough graph on at least three vertices is Hamiltonian. This conjecture has inspired extensive research and has been verified for several special classes…

We prove that, for large $n$, every $3$-connected $D$-regular graph on $n$ vertices with $D \geq n/4$ is Hamiltonian. This is best possible and confirms a conjecture posed independently by Bollob\'as and H\"aggkvist in the 1970s. The proof…

A graph $ G $ is called $ t $-tough if $ \left|S\right|\geq t\cdot w\left(G-S\right)$ for every cutset $ S $ of $G$. Chv\'atal conjectured that there exists a constant $ t_{0} $ such that every $ t_{0} $-tough graph has a hamiltonian cycle.…

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)|…

A set of vertices in a graph is a Hamiltonian subset if it induces a subgraph containing a Hamiltonian cycle. Kim, Liu, Sharifzadeh and Staden proved that among all graphs with minimum degree $d$, $K_{d+1}$ minimises the number of…

Ore's Theorem states that if $G$ is an $n$-vertex graph and every pair of non-adjacent vertices has degree sum at least $n$, then $G$ is Hamiltonian. A $[3]$-graph is a hypergraph in which every edge contains at most $3$ vertices. In this…

In 1973, Chv\'atal conjectured that there exists a constant $t_0$ such that every $t_0$-tough graph on at least three vertices is Hamiltonian. While this conjecture is still open, work has been done to confirm it for several graph classes,…

Here we study some algebraic properties of non-cyclic graphs. In this paper we show that $\overline{\Gamma}_G$ is isomorphic to $K_3\cup (n-4)K_1$ or $K_4\cup (n-5)K_1$ if and only if $G$ is isomorphic to $D_8$ or $D_{10}$, respectively. We…

Let $G$ be an undirected graph of order $n$ and let $C_i$ be an $i$-cycle graph. $G$ is called pancyclic if $G$ contains a $C_i$ for any $i\in \{3,4,\ldots,n\}$. We show that the pancyclicity of specific Cayley graphs and the Cartesian…

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…

We show that every $3$-uniform hypergraph $H=(V,E)$ with $|V(H)|=n$ and minimum pair degree at least $(4/5+o(1))n$ contains a squared Hamiltonian cycle. This may be regarded as a first step towards a hypergraph version of the P\'osa-Seymour…

Let claw be the graph $K_{1,3}$. A graph $G$ on $n\geq 3$ vertices is called \emph{o}-heavy if each induced claw of $G$ has a pair of end-vertices with degree sum at least $n$, and 1-heavy if at least one end-vertex of each induced claw of…

The {\em bipartite-hole-number} of a graph $G$, denoted as $\widetilde\alpha(G)$, is the minimum number $k$ such that there exist integers $a$ and $b$ with $a + b = k+1$ such that for any two disjoint sets $A, B \subseteq V(G)$, there is an…