Related papers: Paths in hypergraphs: a rescaling phenomenon

Given integers $\ell,n$, the $\ell$th power of the path $P_n$ is the ordered graph $P_n^{\ell}$ with vertex set $v_1<v_2<\cdots < v_n$, and all edges of the form $v_iv_j$ where $|i-j|\le \ell$. The ramsey number $r(P_n^{\ell}, P_n^{\ell})$…

A hypergraph $G=(V,E)$ is $(k,\ell)$-sparse if no subset $V'\subset V$ spans more than $k|V'|-\ell$ hyperedges. We characterize $(k,\ell)$-sparse hypergraphs in terms of graph theoretic, matroidal and algorithmic properties. We extend…

A graph is $k$-planar if it can be drawn in the plane such that no edge is crossed more than $k$ times. While for $k=1$, optimal $1$-planar graphs, i.e., those with $n$ vertices and exactly $4n-8$ edges, have been completely characterized,…

A graph $G$ is $k$-critical (list $k$-critical, DP $k$-critical) if $\chi(G)= k$ ($\chi_\ell(G)= k$, $\chi_\mathrm{DP}(G)= k$) and for every proper subgraph $G'$ of $G$, $\chi(G')<k$ ($\chi_\ell(G')< k$, $\chi_\mathrm{DP}(G')<k$). Let $f(n,…

Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is $m$-\textit{coloured} if each of the $m$ colours is used. For an $m$-colouring $\Delta$ of $\mathbb{N}^{(2)}$, the complete graph on $\mathbb{N}$, we…

Let $G=(V,E)$ be a graph of density $p$ on $n$ vertices. Following Erd\H{o}s, \L uczak and Spencer, an $m$-vertex subgraph $H$ of $G$ is called {\em full} if $H$ has minimum degree at least $p(m - 1)$. Let $f(G)$ denote the order of a…

For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem…

We consider the structure of $H$-free subgraphs of graphs with high minimal degree. We prove that for every $k>m$ there exists an $\epsilon:=\epsilon(k,m)>0$ so that the following holds. For every graph $H$ with chromatic number $k$ from…

The transversal rank of a hypergraph is the maximum size of its minimal hitting sets. Deciding, for an $n$-vertex, $m$-edge hypergraph and an integer $k$, whether the transversal rank is at least $k$ takes time $O(m^{k+1} n)$ with an…

We study the emergence of loose Hamilton cycles in subgraphs of random hypergraphs. Our main result states that the minimum $d$-degree threshold for loose Hamiltonicity relative to the random $k$-uniform hypergraph $H_k(n,p)$ coincides with…

For a family of graphs $\mathcal{F}$, the Tur\'{a}n number $ex(n,\mathcal{F})$ is the maximum number of edges in an $n$-vertex graph containing no member of $\mathcal{F}$ as a subgraph. The maximum number of edges in an $n$-vertex connected…

A graph construction that produces a k-regular graph on n vertices for any choice of k >= 3 and n = m(k+1) for integer m >= 2 is described. The number of Hamiltonian cycles in such graphs can be explicitly determined as a function of n and…

A graph on $n$ vertices is called pancyclic if it contains a cycle of every length $3\le l \le n$. Given a Hamiltonian graph $G$ with independence number at most $k$ we are looking for the minimum number of vertices $f(k)$ that guarantees…

We give a short proof that any k-uniform hypergraph H on n vertices with bounded degree \Delta has Ramsey number at most c(\Delta, k)n, for an appropriate constant c(\Delta, k). This result was recently proved by several authors, but those…

Let $K_k$, $C_k$, $T_k$, and $P_{k}$ denote a complete graph on $k$ vertices, a cycle on $k$ vertices, a tree on $k+1$ vertices, and a path on $k+1$ vertices, respectively. Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the…

We prove that, for all $k \ge 3,$ and any integers $\Delta, n$ with $n \ge \Delta,$ there exists a $k$-uniform hypergraph on $n$ vertices with maximum degree at most $\Delta$ whose $4$-color Ramsey number is at least $\mathrm{tw}_k(c_k…

For a 3-uniform hypergraph (3-graph) $F$, let $r(F,n)$ be the smallest $N$ such that any $N$-vertex $F$-free 3-graph has an independent set of size $n$. We construct a $3$-graph $H_2$ with six vertices and five edges such that…

Morris and Saxton used the method of containers to bound the number of $n$-vertex graphs with $m$ edges containing no $\ell$-cycles, and hence graphs of girth more than $\ell$. We consider a generalization to $r$-uniform hypergraphs. The…

For two graphs $F$ and $H$, the relative Tur\'{a}n number $\mathrm{ex}(H,F)$ is the maximum number of edges in an $F$-free subgraph of $H$. Foucaud, Krivelevich, and Perarnau \cite{FKP} and Perarnau and Reed \cite{PR} studied these…

Let $G$ be a directed graph on finitely many vertices and edges, and assign a positive weight to each edge on $G$. Fix vertices $u$ and $v$ and consider the set of paths that start at $u$ and end at $v$, self-intersecting in any number of…