Related papers: Minimum degree ensuring that a hypergraph is hamil…

A conjecture of Carsten Thomassen states that every 4-connected line graph is hamiltonian. It is known that the conjecture is true for 7-connected line graphs. We improve this by showing that any 5-connected line graph of minimum degree at…

Dirac's theorem states that any $n$-vertex graph $G$ with even integer $n$ satisfying $\delta(G) \geq n/2$ contains a perfect matching. We generalize this to $k$-uniform linear hypergraphs by proving the following. Any $n$-vertex…

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

Given a set $R$, a hypergraph is $R$-uniform if the size of every hyperedge belongs to $R$. A hypergraph $\mathcal{H}$ is called \textit{covering} if every vertex pair is contained in some hyperedge in $\mathcal{H}$. In this note, we show…

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…

Recently it was shown (by the author) that every graph of size $q$ (the number of edges) and minimum degree $\delta$ is hamiltonian if $q\le\delta^2+\delta-1$ (arXiv:1107.2201v1). In this paper we present the exact analog of this result for…

We say that a k-uniform hypergraph C is an l-cycle if there exists a cyclic ordering of the vertices of C such that every edge of C consists of k consecutive vertices and such that every pair of consecutive edges (in the natural ordering of…

The Hamiltonian number of a connected graph is the minimum of the lengths of the closed, spanning walks in the graph. In 1968, Grinberg published a necessary condition for the existence of a Hamiltonian cycle in a planar graph, formulated…

A spanning tree with no vertices of degree 2 is called a Homeomorphically irreducible spanning tree\,(HIST). Based on a HIST embedded in the plane, a Halin graph is formed by connecting the leaves of the tree into a cycle following the…

Let $k_r(n,\delta)$ be the minimum number of $r$-cliques in graphs with $n$ vertices and minimum degree $\delta$. We evaluate $k_r(n,\delta)$ for $\delta \leq 4n/5$ and some other cases. Moreover, we give a construction, which we conjecture…

L.H. Feng at el \cite{feng4} present sufficient conditions based on spectral radius for a graph with large minimum degree to be $s$-path-coverable and $s$-Hamiltonian. Motivated by this study, in this paper, we give the sufficient…

We determine the minimum degree sum of two adjacent vertices that ensures a perfect matching in a 3-graph without isolated vertex. More precisely, suppose that $H$ is a 3-uniform hypergraph whose order $n$ is sufficiently large and…

A graph is hypohamiltonian if it is not Hamiltonian, but the deletion of any single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 42 vertices, a result due to Araya and Wiener. That result…

Let $G$ be a $t$-tough graph on $n\ge 3$ vertices for some $t>0$. It was shown by Bauer et al. in 1995 that if the minimum degree of $G$ is greater than $\frac{n}{t+1}-1$, then $G$ is hamiltonian. In terms of Ore-type hamiltonicity…

A graph G on n vertices is Hamiltonian if it contains a cycle of length n and pancyclic if it contains cycles of length $\ell$ for all $3 \le \ell \le n$. Write $\alpha(G)$ for the independence number of $G$, i.e. the size of the largest…

We prove that for $r\in \mathbb{N}$ with $r\geq 2$ and $\mu>0$, there exist $\alpha>0$ and $n_{0}$ such that for every $n\geq n_{0}$, every $n$-vertex graph $G$ with $\delta(G)\geq \left(1-\frac{1}{r}+\mu\right)n$ and $\alpha(G)\leq \alpha…

Let $G$ be a connected graph with minimum degree $\delta(G)$ and vertex-connectivity $\kappa(G)$. The graph $G$ is $k$-connected if $\kappa(G)\geq k$, maximally connected if $\kappa(G) = \delta(G)$, and super-connected (or super-$\kappa$)…

Dirac proved that any graph with minimum vertex degree $\delta$ contains either a cycle of length at least $2\delta$ or a Hamilton cycle. Motivated by this result, we characterize those graphs having no cycle longer than $2\delta$.

The edge-connectivity of a graph is the minimum number of edges whose deletion disconnects the graph. Let $\Delta(G)$ the maximum degree of a graph $G$ and let $\rho(G)$ be the spectral radius of $G$. In this article we present a lower…

We study matching-removability under the degree/connectivity regime of Halin's theorem, which asserts that every $k$-connected graph $G$ with minimum degree $\delta(G)\ge k+1$ contains an edge $e$ such that $G-e$ remains $k$-connected. For…