Related papers: More about sparse halves in triangle-free graphs

Bollob\'as proved that for every $k$ and $\ell$ such that $k\mathbb{Z}+\ell$ contains an even number, an $n$-vertex graph containing no cycle of length $\ell \bmod k$ can contain at most a linear number of edges. The precise (or asymptotic)…

Consider the triangle-free graph process, which starts from the empty graph on $n$ vertices and a random ordering of the possible ${n \choose 2}$ edges; the edges are added in this ordering provided the graph remains triangle free. We will…

We show that the size of a 4-critical graph of girth at least five is bounded by a linear function of its genus. This strengthens the previous bound on the size of such graphs given by Thomassen. It also serves as the basic case for the…

In 1975, P. Erd\"os proposed the problem of determining the maximum number $f(n)$ of edges in a graph of $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $$f(n)\geq…

Given a finite projective plane $\Pi$ and a polarity $\theta$ of $\Pi$, the corresponding polarity graph is the graph whose vertices are the points of $\Pi$. Two distinct vertices $p$ and $p'$ are adjacent if $p$ is incident to $\theta…

In this paper we show that the maximum number of hyperedges in a $3$-uniform hypergraph on $n$ vertices without a (Berge) cycle of length five is less than $(0.254 + o(1))n^{3/2}$, improving an estimate of Bollob\'as and Gy\H{o}ri. We…

It is shown that for a constant $t\in \mathbb{N}$, every simple topological graph on $n$ vertices has $O(n)$ edges if it has no two sets of $t$ edges such that every edge in one set is disjoint from all edges of the other set (i.e., the…

We show that for every integer $n\geq 1$ there exists a graph $G_n$ with $(1+o(1))n$ vertices and $n^{1 + o(1)}$ edges such that every $n$-vertex planar graph is isomorphic to a subgraph of $G_n$. The best previous bound on the number of…

A bridgeless graph $G$ is called $3$-flow-critical if it does not admit a nowhere-zero $3$-flow, but $G/e$ has for any $e\in E(G)$. Tutte's $3$-flow conjecture can be equivalently stated as that every $3$-flow-critical graph contains a…

In 1965 Erd\H{o}s conjectured that the number of edges in k-uniform hypergraphs on n vertices in which the largest matching has s edges is maximized for hypergraphs of one of two special types. We settled this conjecture in the affirmative…

The Petersen colouring conjecture states that every bridgeless cubic graph admits an edge-colouring with $5$ colours such that for every edge $e$, the set of colours assigned to the edges adjacent to $e$ has cardinality either $2$ or $4$,…

One of the oldest results in modern graph theory, due to Mantel, asserts that every triangle-free graphs on $n$ vertices has at most $\lfloor n^2/4\rfloor$ edges. About half a century later Andr\'asfai studied dense triangle-free graphs and…

An intersection graph of curves in the plane is called a string graph. Matousek almost completely settled a conjecture of the authors by showing that every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log m). In the…

The cube Q is the usual 8-vertex graph with 12 edges. Here we give a new proof for a theorem of Erd\H{o}s and Simonovits concerning the Tur\'an number of the cube. Namely, it is shown that e(G) < n^{8/5}+(2n)^{3/2} holds for any n-vertex…

A drawing of a graph in the plane is a thrackle if every pair of edges intersects exactly once, either at a common vertex or at a proper crossing. Conway's conjecture states that a thrackle has at most as many edges as vertices. In this…

Let $\mathrm{ex}(n,s)$ denote the maximum number of edges in a triangle-free graph on $n$ vertices which contains no independent sets larger than $s$. The behaviour of $\mathrm{ex}(n,s)$ was first studied by Andr\'asfai, who conjectured…

A normal 5-edge-coloring of a cubic graph is a coloring such that for every edge the number of distinct colors incident to its end-vertices is 3 or 5 (and not 4). The well known Petersen Coloring Conjecture is equivalent to the statement…

In 1988, Erd\H{o}s suggested the question of minimizing the number of edges in a connected $n$-vertex graph where every edge is contained in a triangle. Shortly after, Catlin, Grossman, Hobbs, and Lai resolved this in a stronger form. In…

We prove a vertex domination conjecture of Erd\H os, Faudree, Gould, Gy\'arf\'as, Rousseau, and Schelp, that for every n-vertex complete graph with edges coloured using three colours there exists a set of at most three vertices which have…

There are finitely many graphs with diameter $2$ and girth 5. What if the girth 5 assumption is relaxed? Apart from stars, are there finitely many triangle-free graphs with diameter $2$ and no $K_{2,3}$ subgraph? This question is related to…