Related papers: A sufficient condition for a hypergraph to have a …

In the language of hypergraphs, our main result is a Dirac-type bound: we prove that every $3$-connected hypergraph $H$ with $ \delta(H)\geq \max\{|V(H)|, \frac{|E(H)|+10}{4}\}$ has a hamiltonian Berge cycle. This is sharp and refines a…

Consider the random process in which the edges of a graph $G$ are added one by one in a random order. A classical result states that if $G$ is the complete graph $K_{2n}$ or the complete bipartite graph $K_{n,n}$, then typically a perfect…

It is well known that if $G = (V, E)$} is a multigraph and $X\subset V$ is a subset of even order, then $G$ contains a spanning forest $H$ such that each vertex from $X$ has an odd degree in $H$ and all the other vertices have an even…

A graph $G$ of order $n$ is said to be $k$-factor-critical for integers $1\leq k< n$, if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$-factor-critical graph is minimal if for every edge, the deletion of…

Testing if a given graph $G$ contains the $k$-vertex path $P_k$ as a minor or as an induced minor is trivial for every fixed integer $k\geq 1$. However, the situation changes for the problem of checking if a graph can be modified into $P_k$…

A $k$-uniform hypergraph $G=(V,E)$ is called odd-bipartite ([5]), if $k$ is even and there exists some proper subset $V_1$ of $V$ such that each edge of $G$ contains odd number of vertices in $V_1$. Odd-bipartite hypergraphs are…

A graph is called an $(r,k)$-graph if its vertex set can be partitioned into $r$ parts, each having at most $k$ vertices and there is at least one edge between any two parts. Let $f(r,H)$ be the minimum $k$ for which there exists an…

A graph $G$ is equimatchable if any matching in $G$ is a subset of a maximum-size matching. It is known that any $2$-connected equimatchable graph is either bipartite or factor-critical. We prove that for any vertex $v$ of a $2$-connected…

A vertex subset $S$ of a graph $G$ is a double dominating set of $G$ if $|N[v]\cap S|\geq 2$ for each vertex $v$ of $G$, where $N[v]$ is the set of the vertex $v$ and vertices adjacent to $v$. The double domination number of $G$, denoted by…

The \textsc{Bipartite Contraction} problem is to decide, given a graph $G$ and a parameter $k$, whether we can can obtain a bipartite graph from $G$ by at most $k$ edge contractions. The fixed-parameter tractability of the problem was shown…

Let $n,k$ be positive integers such that $n\geq k$ and $G$ be a tripartite graph with parts $A,B,C$ such that $|A|=|B|=|C|=n$. Denote the edge densities of $G[A,B]$, $G[A,C]$ and $G[B,C]$ by $\alpha$, $\beta$ and $\gamma$, respectively. In…

Given $k\ge 2$ and two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, an $F$-factor in $H$ is a set of vertex disjoint copies of $F$ that together cover the vertex set of $H$. Lenz and Mubayi were first to study the $F$-factor problems…

Let $\mathscr{B}_n = \{ \pm x_1, \pm x_2, \pm x_3, \cdots, \pm x_{n-1}, x_n \}$ where $n>1$ is fixed, $x_i \in \mathbb{R}^+$, $i = 1, 2, 3, \cdots, n$ and $x_1 < x_2 < x_3 < \cdots < x_n$. Let $\phi(\mathscr{B}_n)$ be the set of all…

Let $H$ be a fixed undirected graph on $k$ vertices. The $H$-hitting set problem asks for deleting a minimum number of vertices from a given graph $G$ in such a way that the resulting graph has no copies of $H$ as a subgraph. This problem…

For a bipartite graph G, let h(G) be the largest t such that either G or the bipartite complement of G contain K_{t,t}. For a class F of graphs, let h(F)= min {h(G): G\in F}. We say that a bipartite graph H is strongly acyclic if neither H…

Let $ G $ be a graph. A subset $S \subseteq V(G) $ is called a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $S$. The total domination number, $\gamma_{t}$($G$), is the minimum cardinality of a total…

We prove inequalities between the densities of various bipartite subgraphs in signed graphs and graphons. One of the main inequalities is that the density of any bipartite graph with girth r cannot exceed the density of the r-cycle. This…

A graph is said to be edge-transitive if its automorphism group acts transitively on its edges. It is known that edge-transitive graphs are either vertex-transitive or bipartite. In this paper we present a complete classification of all…

Let $G$ be a graph with vertex set $V$ and let $g, f : V\rightarrow \mathbb{Z}^+$ be two functions such that $g\le f$. We say that $G$ has all $(g, f )$-factors if $G$ has an $h$-factor for every $h: V\rightarrow \mathbb{Z}^+$ such that…

In this paper, we investigate the problem of finding {\it bisections} (i.e., balanced bipartitions) in graphs. We prove the following two results for {\it all} graphs $G$: (1). $G$ has a bisection where each vertex $v$ has at least $(1/4 -…