Related papers: Asymptotic structure. IV. A counterexample to the …

A celebrated theorem of Stiebitz asserts that any graph with minimum degree at least $s+t+1$ can be partitioned into two parts which induce two subgraphs with minimum degree at least $s$ and $t$, respectively. This resolved a conjecture of…

A graph is "$H$-free" if it has no induced subgraph isomorphic to $H$. A conjecture of Conlon, Fox and Sudakov states that for every graph $H$, there exists $s>0$ such that in every $H$-free graph with $n>1$ vertices, either some vertex has…

We study a generalization of the well-known disjoint paths problem which we call the metric Menger problem, denoted MM(r,k), where one is given two subsets of a graph and must decide whether they can be connected by $k$ paths of pairwise…

We prove that there exists an absolute constant $C>0$ such that, for any positive integer $k$, every graph $G$ with minimum degree at least $Ck$ admits a vertex-partition $V(G)=S\cup T$, where both $G[S]$ and $G[T]$ have minimum degree at…

Let $S$ be a finite non-commutative semigroup. The commuting graph of $S$, denoted $\cg(S)$, is the graph whose vertices are the non-central elements of $S$ and whose edges are the sets $\{a,b\}$ of vertices such that $a\ne b$ and $ab=ba$.…

A famous conjecture by Thomassen from 1983 asserts that for any given $k,g\in \mathbb{N}$ there exists some $d=d(k,g)\in \mathbb{N}$ such that every graph of minimum degree at least $d$ contains a subgraph of minimum degree at least $k$ and…

In 2010, Mkrtchyan, Petrosyan and Vardanyan proved that every graph $G$ with $2\leq \delta(G)\leq \Delta(G)\leq 3$ contains a maximum matching whose unsaturated vertices do not have a common neighbor, where $\Delta(G)$ and $\delta(G)$…

This is a companion paper to the paper "Hyperstability in the Erdos-Sos Conjecture". In that paper the following rough structure theorem was proved for graphs G containing no copy of a bounded degree tree T: from any such G, one can delete…

Thomassen's chord conjecture from 1976 states that every longest cycle in a $3$-connected graph has a chord. This is one of the most important unsolved problems in graph theory. Let $H$ be a subgraph of a graph $G$. A vertex $v$ of $H$ is…

Let $k$ be a positive integer. Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles. It is famous as one of the one hundred unsolved problems selected in…

We study the Decomposition Conjecture posed by Bar\'at and Thomassen (2006), which states that for every tree $T$ there exists a natural number $k_T$ such that, if $G$ is a $k_T$-edge-connected graph and $|E(T)|$ divides $|E(G)|$, then $G$…

One theorem of Nemhauser and Trotter ensures that, under certain conditions, a stable set of a graph G can be enlarged to a maximum stable set of this graph. For example, any stable set consisting of only simplicial vertices is contained in…

We introduce the notion of coarse bottlenecking in graphs and coarse skeletons of graphs and show how bottlenecking guarantees that a skeleton resembles (up to quasi-isometry) the original graph. We show how these tools can be used to…

Let $\mathscr{G}$ be the class of plane graphs without triangles normally adjacent to $8^{-}$-cycles, without $4$-cycles normally adjacent to $6^{-}$-cycles, and without normally adjacent $5$-cycles. In this paper, it is shown that every…

A stable set in a graph G is a set of mutually non-adjacent vertices, alpha(G) is the size of a maximum stable set of G, and core(G) is the intersection of all its maximum stable sets. In this paper we demonstrate that in a tree T, of order…

This paper studies higher index theory for a random sequence of bounded degree, finite graphs with diameter tending to infinity. We show that in a natural model for such random sequences the following hold almost surely: the coarse…

Let $k \ge 3$ be an integer, $H_{k}(G)$ be the set of vertices of degree at least $2k$ in a graph $G$, and $L_{k}(G)$ be the set of vertices of degree at most $2k-2$ in $G$. In 1963, Dirac and Erd\H{o}s proved that $G$ contains $k$…

Thomassen's chord conjecture from 1976 states that every longest cycle in a $3$-connected graph has a chord. The circumference $c(G)$ and induced circumference $c'(G)$ of a graph $G$ are the length of its longest cycles and the length of…

Let $c:V\cup E\to\{1,2,\ldots,k\}$ be a (not necessarily proper) total colouring of a graph $G=(V,E)$ with maximum degree $\Delta$. Two vertices $u,v\in V$ are sum distinguished if they differ with respect to sums of their incident colours,…

Kriesel conjectured that every minimally $1$-tough graph has a vertex with degree precisely $2$. Katona and Varga (2018) proposed a generalized version of this conjecture which says that every minimally $t$-tough graph has a vertex with…