Related papers: Complete minors and average degree -- a short proo…

We show that any self-complementary graph with $n$ vertices contains a $K_{\lfloor \frac{n+1}{2}\rfloor}$ minor. We derive topological properties of self-complementary graphs.

We prove that every locally finite, quasi-transitive graph with a thick end whose cycle space is generated by cycles of bounded length contains the full-grid as an asymptotic minor and as a diverging minor. This in particular includes all…

Much of extremal graph theory has concentrated either on finding very small subgraphs of a large graph (Turan-type results) or on finding spanning subgraphs (Dirac-type results). In this paper we are interested in finding intermediate-sized…

The celebrated Andr\'{a}sfai--Erd\H{o}s--S\'{o}s Theorem from 1974 shows that every $n$-vertex triangle-free graph with minimum degree greater than $2n/5$ must be bipartite. We establish a positive codegree extension of this result for the…

We propose a randomized algorithm with query access that given a graph $G$ with arboricity $\alpha$, and average degree $d$, makes $\widetilde{O}\left(\frac{\alpha}{\varepsilon^2d}\right)$ \texttt{Degree} and…

Motivated by a conjecture of Gy\'arf\'as, recently B\"ottcher, Hladk\'y, Piguet, and Taraz showed that every collection $T_1,\dots,T_t$ of trees on $n$ vertices with $\sum_{i=1}^te(T_i)\leq \binom{n}{2}$ and with bounded maximum degree, can…

A subset of vertices in a graph is called a total dominating set if every vertex of the graph is adjacent to at least one vertex of this set. A total dominating set is called minimal if it does not properly contain another total dominating…

We show that every proper minor-closed class of graphs admits a $(1+o(1))\log_2 n$-bit adjacency labelling scheme. Equivalently, for every proper minor-closed class $\mathcal{G}$ and every positive integer $n$ there exists an…

In this paper we show that every graph of pathwidth less than $k$ that has a path of order $n$ also has an induced path of order at least $\frac{1}{3} n^{1/k}$. This is an exponential improvement and a generalization of the polylogarithmic…

We prove that if the minors of degree $k$ of a Sobolev map $\mathbb{R}^d \to \mathbb{R}^d$ are smooth then the map is smooth, when $k,d$ are not both even. We use this result to derive a simple, self-contained proof of the famous Liouville…

Hajnal and Szemeredi proved that every graph G with |G|=ks and minimum degree at least k(s-1) contains k vertex disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every…

We prove the extremal function for $K_9^=$ minors, where $K_9^=$ denotes the complete graph $K_9$ with two edges removed. In particular, we show that any graph with $n$ vertices and at least $6n - 20$ edges either contains a $K_9^=$ minor…

A recent paper of Balogh, Li and Treglown initiated the study of Dirac-type problems for ordered graphs. In this paper we prove a number of results in this area. In particular, we determine asymptotically the minimum degree threshold for…

Our main result improves bounds of Markstrom and Rucinski on the minimum d-degree which forces a perfect matching in a k-uniform hypergraph on n vertices. We also extend bounds of Bollobas, Daykin and Erdos by asymptotically determining the…

The set D of distinct signed degrees of the vertices in a signed graph G is called its signed degree set. In this paper, we prove that every non-empty set of positive (negative) integers is the signed degree set of some connected signed…

In this paper, we establish a couple of results on extremal problems in bipartite graphs. Firstly, we show that every sufficiently large bipartite graph with average degree $D$ and with $n$ vertices on each side has a balanced independent…

A connected $k$-chromatic graph $G$ is said to be {\it double-critical} if for all edges $uv$ of $G$ the graph $G - u - v$ is $(k-2)$-colourable. A longstanding conjecture of Erd\H{o}s and Lov\'asz states that the complete graphs are the…

Let $T$ be an oriented tree on $n$ vertices with maximum degree at most $e^{o(\sqrt{\log n})}$. If $G$ is a digraph on $n$ vertices with minimum semidegree $\delta^0(G)\geq(\frac12+o(1))n$, then $G$ contains $T$ as a spanning tree, as…

Perhaps the very first elementary exercise one encounters in graph theory is the result that any graph on at least two vertices must have at least two vertices with the same degree. There are various ways in which this result can be…

We prove that every graph which admits a tree-decomposition into finite parts has a rooted tree-decomposition into finite parts that is linked, tight and componental. As an application, we obtain that every graph without half-grid minor has…