Related papers: A degree sum condition for hamiltonicity in balanc…

Given a graph $H$, a graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. Shi and Shan conjectured that every $1$-tough $2k$-connected $(P_2 \cup kP_1)$-free graph is hamiltonian for $k \geq 4$. This conjecture has been…

A $k$-edge-coloured graph is colour-balanced if each colour appears equally often. Resolving a conjecture of Pardey and Rautenbach, we show that any colour-balanced $k$-edge-coloured complete graph $K_{2kt}$ contains a perfect matching that…

For a connected graph, the Hamiltonian cycle (path) is a simple cycle (path) that spans all the vertices in the graph. It is known from \cite{muller,garey} that HAMILTONIAN CYCLE (PATH) are NP-complete in general graphs and chordal…

In this paper, a theorem is proved that generalizes several existing amalgamation results in various ways. The main aim is to disentangle a given edge-colored amalgamated graph so that the result is a graph in which the edges are shared out…

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…

We investigate the terminal-pairibility problem in the case when the base graph is a complete bipartite graph, and the demand graph is also bipartite with the same color classes. We improve the lower bound on maximum value of $\Delta(D)$…

For a graph $G$, the $t$-th power $G^t$ is the graph on $V(G)$ such that two vertices are adjacent if and only if they have distance at most $t$ in $G$; and the $t$-th bi-power $G_B^t$ is the graph on $V(G)$ such that two vertices are…

We construct infinitely many connected, circulant digraphs of outdegree three that have no hamiltonian circuit. All of our examples have an even number of vertices, and our examples are of two types: either every vertex in the digraph is…

Bipartite graphs are a fundamental concept in graph theory with diverse applications. A graph is bipartite iff it contains no odd cycles, a characteristic that has many implications in diverse fields ranging from matching problems to the…

Chen, Faudree, Gould, Jacobson, and Lesniak determined the minimum degree threshold for which a balanced $k$-partite graph has a Hamiltonian cycle. We give an asymptotically tight minimum degree condition for Hamiltonian cycles in arbitrary…

We call a bipartite graph {\it homogeneous} if every finite partial automorphism which respects left and right can be extended to a total automorphism. A $(\kappa,{\lambda} )$ bipartite graph is a bipartite graph with left side of size…

Let $G$ be an $n$-vertex graph with $n\ge 3$. A classic result of Dirac from 1952 asserts that $G$ is hamiltonian if $\delta(G)\ge n/2$. Dirac's theorem is one of the most influential results in the study of hamiltonicity and by now there…

A bicirculant is a regular, $d$-valent graph that admits a semiregular automorphism of order $m$ having two vertex-orbits of size $m$. The vertices of each orbit induce a circulant graph of order $m$ and the remaining edges span a regular…

A graph $G$ is $\ell$-distance-balanced if for each pair of vertices $x$ and $y$ at distance $\ell$ in $G$, the number of vertices closer to $x$ than to $y$ is equal to the number of vertices closer to $y$ than to $x$. A complete…

In a graph $G$, the $2$-neighborhood of a vertex set $X$ consists of all vertices of $G$ having at least $2$ neighbors in $X$. We say that a bipartite graph $G(A,B)$ satisfies the double Hall property if $|A|\geq2$, and every subset $X…

Let $t>0$ be a real number and $G$ be a graph. We say $G$ is $t$-tough if for every cutset $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. The Toughness Conjecture of Chv\'atal, stating that there exists…

Let $D$ be a strongly connected digraphs on $n\ge 4$ vertices. A vertex $v$ of $D$ is noncritical, if the digraph $D-v$ is strongly connected. We prove, that if sum of the degrees of any two adjacent vertices of $D$ is at least $n+1$, then…

We study $M$-alternating Hamilton paths and $M$-alternating Hamilton cycles in a simple connected graph $G$ on $\nu$ vertices with a perfect matching $M$. Let $G$ be a bipartite graph, we prove that if for any two vertices $x$ and $y$ in…

It is well known that a graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric. In the present paper, this assertion is dissected into three separate matrix results of wider scope, which are extended also to…

Bedrossian characterized all pairs of forbidden subgraphs for a 2-connected graph to be Hamiltonian. Instead of forbidding some induced subgraphs, we relax the conditions for graphs to be Hamiltonian by restricting Ore- and Fan-type degree…