Related papers: Supersaturation of odd linear cycles

An ${\ell}$-cycle system ${\mathcal F}$ of a graph $\Gamma$ is a set of ${\ell}$-cycles which partition the edge set of $\Gamma$. Two such cycle systems ${\mathcal F}$ and ${\mathcal F}'$ are said to be {\em orthogonal} if no two distinct…

We prove that every properly edge-colored $n$-vertex graph with average degree at least $100(\log n)^2$ contains a rainbow cycle, improving upon $(\log n)^{2+o(1)}$ bound due to Tomon. We also prove that every properly colored $n$-vertex…

Let $\mc{F}$ be a family of graphs. A graph is {\em $\mc{F}$-free} if it contains no copy of a graph in $\mc{F}$ as a subgraph. A cornerstone of extremal graph theory is the study of the {\em Tur\'an number} $ex(n,\mc{F})$, the maximum…

For any uniformity $r$ and residue $k$ modulo $r$, we give an exact characterization of the $r$-uniform hypergraphs that homomorphically avoid tight cycles of length $k$ modulo $r$, in terms of colorings of $(r-1)$-tuples of vertices. This…

The Kohayakawa-Nagle-R\"odl-Schacht conjecture roughly states that every sufficiently large locally $d$-dense graph $G$ on $n$ vertices must contain at least $(1-o(1))d^{|E(H)|}n^{|V(H)|}$ copies of a fixed graph $H$. Despite its important…

We prove the well-known Brown-Erd\H{o}s-S\'os Conjecture for hypergraphs of large uniformity in the following form: any dense linear $r$-graph $G$ has $k$ edges spanning at most $(r-2)k+3$ vertices, provided the uniformity $r$ of $G$ is…

Let $c$ be an edge-colouring of a graph $G$ such that for every vertex $v$ there are at least $d \ge 2$ different colours on edges incident to $v$. We prove that $G$ contains a properly coloured path of length 2d or a properly coloured…

An induced packing of cycles in a graph is a set of vertex-disjoint cycles with no edges between them. We generalise the classic Erd\H{o}s-P\'osa theorem to induced packings of cycles. More specifically, we show that there exist functions…

In this paper, we study discrepancy questions for spanning subgraphs of $k$-uniform hypergraphs. Our main result is that, for any integers $k \ge 3$ and $r \ge 2$, any $r$-colouring of the edges of a $k$-uniform $n$-vertex hypergraph $G$…

In this paper, we study the maximum number of edges in an $N$-vertex $r$-uniform hypergraph with girth $g$ where $g \in \{5,6 \}$. Writing $\textrm{ex}_r ( N, \mathcal{C}_{<g} )$ for this maximum, it is shown that $\textrm{ex}_r ( N ,…

We describe a general purpose algorithm for counting simple cycles and simple paths of any length $\ell$ on a (weighted di)graph on $N$ vertices and $M$ edges, achieving a time complexity of $O\left(N+M+\big(\ell^\omega+\ell\Delta\big)…

In this short note we determine the maximum number, over all $n$-vertex graphs $G$, of orientations of $G$ containing no strongly connected cycle $C_{2k+1}$. This answers a part of a recent question of Araujo, Botler and Mota.

As the counterpart of classical theorems on cycles of consecutive lengths due to Bondy and Bollob\'as in spectral graph theory, Nikiforov proposed the following open problem in 2008: What is the maximum $C$ such that for all positive…

Let $r \ge 3$ be fixed and $G$ be an $n$-vertex graph. A long-standing conjecture of Gy\H{o}ri states that if $e(G) = t_{r-1}(n) + k$, where $t_{r-1}(n)$ denotes the number of edges of the Tur\'{a}n graph on $n$ vertices and $r - 1$ parts,…

The Erd\H{o}s--Gallai Theorem states that for $k\geq 3$ every graph on $n$ vertices with more than $\frac{1}{2}(k-1)(n-1)$ edges contains a cycle of length at least $k$. Kopylov proved a strengthening of this result for 2-connected graphs…

For fixed positive integers $r, k$ and $\ell$ with $1 \leq \ell < r$ and an $r$-uniform hypergraph $H$, let $\kappa (H, k,\ell)$ denote the number of $k$-colorings of the set of hyperedges of $H$ for which any two hyperedges in the same…

Extremal problems on the $4$-cycle $C_4$ played a heuristic important role in the development of extremal graph theory. A fundamental theorem of F\"uredi states that the Tur\'an number $ex(q^2+q+1, C_4)\leq \frac12 q(q+1)^2$ holds for every…

How many edges can a quadrilateral-free subgraph of a hypercube have? This question was raised by Paul Erd\H{o}s about $27$ years ago. His conjecture that such a subgraph asymptotically has at most half the edges of a hypercube is still…

An $n$-vertex $k$-uniform hypergraph $G$ is $(d,\alpha)$-degenerate if $m_1(G)\le{d}$ and there exists a constant $\varepsilon >0$ such that for every subset $U\subseteq{V(G)}$ with size $2\le|U|\le{\varepsilon n}$, we have…

Let $H_{n,p,r}^{(k)}$ denote a randomly colored random hypergraph, constructed on the vertex set $[n]$ by taking each $k$-tuple independently with probability $p$, and then independently coloring it with a random color from the set $[r]$.…