# Search results for: Jacques Verstraëte

Journal of Graph Theory > 100 > 3 > 543 - 558

*girth*of a hypergraph $H$ is the minimum $\ell \ge 2$ such that there exist distinct vertices ${v}_{1},\text{\u2026},{v}_{\ell}$ and hyperedges ${e}_{1},\text{\u2026},{e}_{\ell}$ with ${v}_{i},{v}_{i+1}\in {e}_{i}$ for all $1\le i$...

Random Structures & Algorithms > 59 > 1 > 79 - 95

*r*‐uniform

*d*‐regular hypergraphs

*G*on

*n*vertices with girth

*g*. By analyzing the expected size of the independent sets generated by this algorithm, we show that $\alpha \left(G\right)\ge \left(f\right(d,r)-\u03f5(g,d,r\left)\right)n$, where $\u03f5(g,d,r)$ converges to 0 as

*g*→

*∞*for fixed

*d*and

*r*, and

*f*(

*d*,

*r*) is determined by a differential equation. This extends earlier results...

Journal of Graph Theory > 88 > 3 > 411 - 427

*p*on

*n*vertices. Following Erdős, Łuczak, and Spencer, an

*m*‐vertex subgraph

*H*of

*G*is called

*full*if

*H*has minimum degree at least $p(m-1)$. Let $f\left(G\right)$ denote the order of a largest full subgraph of

*G*. If $p\left(\frac{n}{2}\right)$ is a nonnegative integer, define Erdős, Łuczak, and Spencer proved that for $n\ge 2$, In this article, we prove the following lower bound: for ${n}^{-\frac{2}{3}}<{p}_{n}<1-{n}^{-\frac{1}{7}}$, Furthermore,...

Journal of Combinatorial Theory, Series B > 2017 > 122 > C > 457-478

Journal of Combinatorial Theory, Series B > 2016 > 121 > C > 197-228

Combinatorica > 2017 > 37 > 3 > 481-494

*G*of chromatic number

*k*≥

*k*

_{0}(

*ε*) contains cycles of at least

*k*

^{2−ε}different lengths as

*k*→∞. In this paper, we prove the stronger fact that every triangle-free graph

*G*of chromatic number

*k*≥

*k*

_{0}(

*ε*) contains cycles of 1/64(1 −

*ε*)

*k*

^{2}log

*k*/4 consecutive lengths, and a cycle of length at least 1/4(1 −

*ε*)

*k*

^{2}log

*k*. As there exist triangle-free...

*G*and we have to find a largest set of edge-disjoint cycles in

*G*. The problem of packing vertex-disjoint cycles in

*G*is defined similarly. The best approximation algorithms for edge-disjoint cycle packing are due to Krivelevich et al. [16], where they give an -approximation for undirected graphs and an $O(\sqrt{n})$...

European Journal of Combinatorics > 2016 > 51 > C > 268-274

Discrete Applied Mathematics > 2015 > 193 > C > 94-101

Discrete Mathematics > 2015 > 338 > 6 > 1000-1010

European Journal of Combinatorics > 2015 > 44 > Part A > 77-86

Journal of Combinatorial Theory, Series A > 2015 > 129 > Complete > 57-79

Journal of Combinatorial Theory, Series B > 2014 > 106 > Complete > 134-162

Random Structures & Algorithms > 44 > 2 > 224 - 239

*independence number*$\alpha \left(H\right)$ of a hypergraph

*H*is the size of a largest set of vertices containing no edge of

*H*. In this paper, we prove that if

*H*

_{n}is an

*n*‐vertex $(r+1)$‐uniform hypergraph in which every

*r*‐element set is contained in at most

*d*edges, where $0<d<n/{\left(\mathrm{log}n\right)}^{3{r}^{2}}$, then where ${c}_{r}>0$ satisfies ${c}_{r}\sim r/e$ as $r\to \infty $. The value of

*c*

_{r}improves and generalizes several earlier results that all use...

Journal of Combinatorial Theory, Series A > 2013 > 120 > 7 > 1491-1507

Combinatorica > 2013 > 33 > 6 > 699-732

*F*be a family of graphs. A graph is

*F-free*if it contains no copy of a graph in

*F*as a subgraph. A cornerstone of extremal graph theory is the study of the

*Turán number*ex(

*n,F)*, the maximum number of edges in an

*F*-free graph on

*n*vertices. Define the

*Zarankiewicz number*z(

*n,F*) to be the maximum number of edges in an

*F*-free

*bipartite*graph on

*n*vertices. Let

*C*

_{ k }denote a cycle of length

*k*, and...

Designs, Codes and Cryptography > 2012 > 65 > 3 > 233-245

*n*≥

*r*≥ 2. A clique partition of $${{[n] \choose r}}$$ is a collection of proper subsets $${A_1, A_2, \ldots, A_t \subset [n]}$$ such that $${\bigcup_i{A_i \choose r}}$$ is a partition of $${{[n]\choose r}}$$ . Let cp(

*n*,

*r*) denote the minimum size of a clique partition of $${{[n] \choose r}}$$ . A classical theorem of de Bruijn and Erdős states that cp(

*n*, 2) = ...

Combinatorica > 2011 > 31 > 5 > 565-581

*k*-core of a graph is the largest subgraph of minimum degree at least

*k*. We show that for

*k*sufficiently large, the threshold for the appearance of a

*k*-regular subgraph in the Erdős-Rényi random graph model

*G*(

*n,p*) is at most the threshold for the appearance of a nonempty (

*k*+2)-core. In particular, this pins down the point of appearance of a

*k*-regular subgraph to a window for

*p*of width roughly...

Journal of Combinatorial Theory, Series A > 2009 > 116 > 7 > 1232-1234

Journal of Combinatorial Theory, Series B > 2009 > 99 > 3 > 643-655