Related papers: On the structure of dense graphs with fixed clique…

Given a graph $G$ and an integer $\ell\ge 2$, we denote by $\alpha_{\ell}(G)$ the maximum size of a $K_{\ell}$-free subset of vertices in $V(G)$. A recent question of Nenadov and Pehova asks for determining the best possible minimum degree…

We prove that for fixed $k$, every $k$-uniform hypergraph on $n$ vertices and of minimum codegree at least $n/2+o(n)$ contains every spanning tight $k$-tree of bounded vertex degree as a sub\-graph. This generalises a well-known result of…

A theorem of R\"odl states that for every fixed $F$ and $\varepsilon>0$ there is $\delta=\delta_F(\varepsilon)$ so that every induced $F$-free graph contains a vertex set of size $\delta n$ whose edge density is either at most $\varepsilon$…

For a graph $G$, let $c_k(G)$ be the number of spanning trees of $G$ with maximum degree at most $k$. For $k \ge 3$, it is proved that every connected $n$-vertex $r$-regular graph $G$ with $r \ge \frac{n}{k+1}$ satisfies $$ c_k(G)^{1/n} \ge…

Two central topics of study in combinatorics are the so-called evolution of random graphs, introduced by the seminal work of Erd\H{o}s and R\'enyi, and the family of $H$-free graphs, that is, graphs which do not contain a subgraph…

A well known theorem in graph theory states that every graph $G$ on $n$ vertices and minimum degree at least $d$ contains a path of length at least $d$, and if $G$ is connected and $n\ge 2d+1$ then $G$ contains a path of length at least…

We give a minimum degree condition sufficent to ensure the existence of a fractional $K_r$-decomposition in a balanced $r$-partite graph (subject to some further simple necessary conditions). This generalises the non-partite problem studied…

A graph $ G $ is minimally $ t $-tough if the toughness of $ G $ is $ t $ and deletion of any edge from $ G $ decreases its toughness. Katona et al. conjectured that the minimum degree of any minimally $ t $-tough graph is $ \lceil 2t\rceil…

We show that if $G$ is a simple triangle-free graph with $n\geq 3$ vertices, without a perfect matching, and having a minimum degree at least $\frac{n-1}{2}$, then $G$ is isomorphic either to $C_5$ or to $K_{\frac{n-1}{2},\frac{n+1}{2}}$.

This paper concerns fractional $K_s$-decompositions of multipartite graphs. For integers $r\ge s\ge 3$, we consider balanced $r$-partite graphs $G$ on $rn$ vertices. We establish necessary conditions for $G$ to admit a fractional…

Let $G$ be a graph. For $x\in V(G)$, let $N(x)=\{y\in V(G)\colon xy\in E(G)\}$. The minimum common degree of $G$, denoted by $\delta_{2}(G)$, is defined as the minimum of $|N(x)\cap N(y)|$ over all non-edges $xy$ of $G$. In 1982,…

Let $\mathcal{H}$ be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let $G$ be an arbitrary finite $\mathcal{H}$-free graph with minimum degree at least $k$. For $p \in [0,1]$, we form a $p$-random…

An immersion of a graph $H$ into a graph $G$ is a one-to-one mapping $f:V(H) \to V(G)$ and a collection of edge-disjoint paths in $G$, one for each edge of $H$, such that the path $P_{uv}$ corresponding to edge $uv$ has endpoints $f(u)$ and…

Erd\H{o}s, Pach, Pollack and Tuza [J. Combin. Theory, B 47, (1989), 279-285] conjectured that the diameter of a $K_{2r}$-free connected graph of order $n$ and minimum degree $\delta\geq 2$ is at most $\frac{2(r-1)(3r+2)}{(2r^2-1)}\cdot…

Let $G$ be a simple connected graph with order $ n(G)$, local metric dimension $ {\rm dim}_l(G)$, local adjacency metric dimension $ {\rm dim}_{A,l}(G)$, and clique number $ \omega(G)$, where $G\not\cong K_{n(G)}$ and $\omega(G)\geq3$. It…

The following question was proposed by Nenadov and Pehova and reiterated by Knierim and Su: Given integers $\ell,r$ and $n$ with $n\in r\mathbb{N}$, is it true that every $n$-vertex graph $G$ with $\delta(G) \ge \max \{ \frac{1}{2},\frac{r…

The smallest number of cliques, covering all edges of a graph $ G $, is called the (edge) clique cover number of $ G $ and is denoted by $ cc(G) $. It is an easy observation that for every line graph $ G $ with $ n $ vertices, $cc(G)\leq n…

We show that for sufficiently large $d$ and for $t\geq d+1$, there is a graph $G$ with average degree $(1-\varepsilon)\lambda t \sqrt{\ln d}$ such that almost every graph $H$ with $t$ vertices and average degree $d$ is not a minor of $G$,…

We show that there is a constant $C$ such that for every $\varepsilon>0$ any $2$-coloured $K_n$ with minimum degree at least $n/4+\varepsilon n$ in both colours contains a complete subgraph on $2t$ vertices where one colour class forms a…

We prove a `resilience' version of Dirac's theorem in the setting of random regular graphs. More precisely, we show that, whenever $d$ is sufficiently large compared to $\varepsilon>0$, a.a.s. the following holds: let $G'$ be any subgraph…