arXiv subjects

Kevin Hendrey

Publications and source records attributed to Kevin Hendrey.

At least 37 records · Page 2Linked to original sources

arXiv2024

Clustered Colouring of Graph Products

A colouring of a graph $G$ has clustering $k$ if the maximum number of vertices in a monochromatic component equals $k$. Motivated by recent results showing that many natural graph classes are subgraphs of the strong product of a graph with bounded treewidth and a path, this paper studies clustered colouring of strong products of two bounded treewidth graphs, where none, one, or both graphs have bounded degree. For example, in the case of two colours, if $n$ is the number of vertices in the product, then we show that clustering $Θ(n^{2/3})$ is best possible, even if one of the graphs is a path. However, if both graphs have bounded degree, then clustering $Θ(n^{1/2})$ is best possible. With three colours, if one of the graphs has bounded degree, then we show that clustering $Θ(n^{3/7})$ is best possible. However, if neither graph has bounded degree, then clustering $Ω(n^{1/2})$ is necessary. More general bounds for any given number of colours are also presented.

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arXiv2024

Optimal bounds for zero-sum cycles. I. Odd order

For a finite (not necessarily Abelian) group $(Γ,\cdot)$, let $n(Γ) \in \mathbb{N}$ denote the smallest positive integer $n$ such that for every labelling of the arcs of the complete digraph of order $n$ using elements from $Γ$, there exists a directed cycle such that the arc-labels along the cycle multiply to the identity. Alon and Krivelevich initiated the study of the parameter $n(\cdot)$ on cyclic groups and proved $n(\mathbb{Z}_q)=O(q \log q)$. This was later improved to a linear bound of $n(Γ)\le 8|Γ|$ for every finite Abelian group by Mészáros and the last author, and then further to $n(Γ)\le 2|Γ|-1$ for every non-trivial finite group independently by Berendsohn, Boyadzhiyska and Kozma as well as by Akrami, Alon, Chaudhury, Garg, Mehlhorn and Mehta. In this series of two papers we conclude this line of research by proving that $n(Γ)\le |Γ|+1$ for every finite group $(Γ,\cdot)$, which is the best possible such bound in terms of the group order and precisely determines the value of $n(Γ)$ for all cyclic groups as $n(\mathbb{Z}_q)=q+1$. In the present paper we prove the above result for all groups of odd order. The proof for groups of even order needs to overcome substantial additional obstacles and will be presented in the second part of this series.

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arXiv2024

When $t$-intersecting hypergraphs admit bounded $c$-strong colourings

The $c$-strong chromatic number of a hypergraph is the smallest number of colours needed to colour its vertices so that every edge sees at least $c$ colours or is rainbow. We show that every $t$-intersecting hypergraph has bounded $(t + 1)$-strong chromatic number, resolving a problem of Blais, Weinstein and Yoshida. In fact, we characterise when a $t$-intersecting hypergraph has large $c$-strong chromatic number for $c\geq t+2$. Our characterisation also applies to hypergraphs which exclude sunflowers with specified parameters.

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arXiv2024

Twin-width of subdivisions of multigraphs

For each $d\leq3$, we construct a finite set $F_d$ of multigraphs such that for each graph $H$ of girth at least $5$ obtained from a multigraph $G$ by subdividing each edge at least two times, $H$ has twin-width at most $d$ if and only if $G$ has no minor in $F_d$. This answers a question of Bergé, Bonnet, and Déprés asking for the structure of graphs $G$ such that each long subdivision of $G$ has twin-width $4$. As a corollary, we show that the $7\times7$ grid has twin-width $4$, which answers a question of Schidler and Szeider.

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arXiv2024

Twin-width of random graphs

We investigate the twin-width of the Erdős-Rényi random graph $G(n,p)$. We unveil a surprising behavior of this parameter by showing the existence of a constant $p^*\approx 0.4$ such that with high probability, when $p^*\le p\le 1-p^*$, the twin-width is asymptotically $2p(1-p)n$, whereas, when $0 p>1-p^*$, the twin-width is significantly higher than $2p(1-p)n$. In addition, we show that the twin-width of $G(n,1/2)$ is concentrated around $n/2 - \sqrt{3n \log n}/2$ within an interval of length $o(\sqrt{n\log n})$. For the sparse random graph, we show that with high probability, the twin-width of $G(n,p)$ is $Θ(n\sqrt{p})$ when $(726\ln n)/n\leq p\leq1/2$.

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arXiv2024

Rainbow saturation for complete graphs

We call an edge-colored graph rainbow if all of its edges receive distinct colors. An edge-colored graph $Γ$ is called $H$-rainbow saturated if $Γ$ does not contain a rainbow copy of $H$ and adding an edge of any color to $Γ$ creates a rainbow copy of $H$. The rainbow saturation number $sat(n,{R}(H))$ is the minimum number of edges in an $n$-vertex $H$-rainbow saturated graph. Girão, Lewis, and Popielarz conjectured that $sat(n,{R}(K_r))=2(r-2)n+O(1)$ for fixed $r\geq 3$. Disproving this conjecture, we establish that for every $r\geq 3$, there exists a constant $α_r$ such that $$r + Ω\left(r^{1/3}\right) \le α_r \le r + r^{1/2} \qquad \text{and} \qquad sat(n,{R}(K_r)) = α_r n + O(1).$$ Recently, Behague, Johnston, Letzter, Morrison, and Ogden independently gave a slightly weaker upper bound which was sufficient to disprove the conjecture. They also introduced the weak rainbow saturation number, and asked whether this is equal to the rainbow saturation number of $K_r$, since the standard weak saturation number of complete graphs equals the standard saturation number. Surprisingly, our lower bound separates the rainbow saturation number from the weak rainbow saturation number, answering this question in the negative. The existence of the constant $α_r$ resolves another of their questions in the affirmative for complete graphs. Furthermore, we show that the conjecture of Girão, Lewis, and Popielarz is true if we have an additional assumption that the edge-colored $K_r$-rainbow saturated graph must be rainbow. As an ingredient of the proof, we study graphs which are $K_r$-saturated with respect to the operation of deleting one edge and adding two edges.

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arXiv2024

Linear bounds on treewidth in terms of excluded planar minors

One of the fundamental results in graph minor theory is that for every planar graph $H$, there is a minimum integer $f(H)$ such that graphs with no minor isomorphic to $H$ have treewidth at most $f(H)$. A lower bound for ${f(H)}$ can be obtained by considering the maximum integer $k$ such that $H$ contains $k$ vertex-disjoint cycles. There exists a graph of treewidth ${Ω(k\log k)}$ which does not contain $k$ vertex-disjoint cycles, from which it follows that ${f(H) = Ω(k\log k)}$. In particular, if ${f(H)}$ is linear in ${\lvert{V(H)}\rvert}$ for graphs $H$ from a subclass of planar graphs, it is necessary that $n$-vertex graphs from the class contain at most ${O(n/\log(n))}$ vertex-disjoint cycles. We ask whether this is also a sufficient condition, and demonstrate that this is true for classes of planar graphs with bounded component size. For an $n$-vertex graph $H$ which is a disjoint union of $r$ cycles, we show that ${f(H) \leq 3n/2 + O(r^2 \log r)}$, and improve this to ${f(H) \leq n + O(\sqrt{n})}$ when ${r = 2}$. In particular this bound is linear when ${r=O(\sqrt{n}/\log(n))}$. We present a linear bound for ${f(H)}$ when $H$ is a subdivision of an $r$-edge planar graph for any constant $r$. We also improve the best known bounds for ${f(H)}$ when $H$ is the wheel graph or the ${4 \times 4}$ grid, obtaining a bound of $160$ for the latter.

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arXiv2023

Twin-width of sparse random graphs

We show that the twin-width of every $n$-vertex $d$-regular graph is at most $n^{\frac{d-2}{2d-2}+o(1)}$ and that almost all $d$-regular graphs attain this bound. More generally, we obtain bounds on the twin-width of sparse Erdős-Renyi and regular random graphs, complementing the bounds in the denser regime due to Ahn, Chakraborti, Hendrey, Kim and Oum.

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arXiv2023

Chromatic number is not tournament-local

Scott and Seymour conjectured the existence of a function $f \colon \mathbb{N} \to \mathbb{N}$ such that, for every graph $G$ and tournament $T$ on the same vertex set, $χ(G) \geqslant f(k)$ implies that $χ(G[N_T^+(v)]) \geqslant k$ for some vertex $v$. In this note we disprove this conjecture even if $v$ is replaced by a vertex set of size $\mathcal{O}(\log{\lvert V(G)\rvert})$. As a consequence, we answer in the negative a question of Harutyunyan, Le, Thomassé, and Wu concerning the corresponding statement where the graph $G$ is replaced by another tournament, and disprove a related conjecture of Nguyen, Scott, and Seymour. We also show that the setting where chromatic number is replaced by degeneracy exhibits a quite different behaviour.

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arXiv2023

On an induced version of Menger's theorem

We prove Menger-type results in which the obtained paths are pairwise non-adjacent, both for graphs of bounded maximum degree and, more generally, for graphs excluding a topological minor. We further show better bounds in the subcubic case, and in particular obtain a tight result for two paths using a computer-assisted proof.

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arXiv2023

Finding dense minors using average degree

Motivated by Hadwiger's conjecture, we study the problem of finding the densest possible $t$-vertex minor in graphs of average degree at least $t-1$. We show that if $G$ has average degree at least $t-1$, it contains a minor on $t$ vertices with at least $(\sqrt{2}-1-o(1))\binom{t}{2}$ edges. We show that this cannot be improved beyond $\left(\frac{3}{4}+o(1)\right)\binom{t}{2}$. Finally, for $t\leq 6$ we exactly determine the number of edges we are guaranteed to find in the densest $t$-vertex minor in graphs of average degree at least $t-1$.

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arXiv2023

Maximal $3$-wise intersecting families

A family $\mathcal{F}$ on ground set $[n]:=\{1,2,\ldots, n\}$ is maximal $k$-wise intersecting if every collection of at most $k$ sets in $\mathcal{F}$ has non-empty intersection, and no other set can be added to $\mathcal{F}$ while maintaining this property. In 1974, Erdős and Kleitman asked for the minimum size of a maximal $k$-wise intersecting family. We answer their question for $k=3$ and sufficiently large $n$. We show that the unique minimum family is obtained by partitioning the ground set $[n]$ into two sets $A$ and $B$ with almost equal sizes and taking the family consisting of all the proper supersets of $A$ and of $B$.

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arXiv2022

Graphs of Linear Growth have Bounded Treewidth

A graph class $\mathcal{G}$ has linear growth if, for each graph $G \in \mathcal{G}$ and every positive integer $r$, every subgraph of $G$ with radius at most $r$ contains $O(r)$ vertices. In this paper, we show that every graph class with linear growth has bounded treewidth.

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arXiv2022

Extremal functions for sparse minors

The "extremal function" $c(H)$ of a graph $H$ is the supremum of densities of graphs not containing $H$ as a minor, where the "density" of a graph $G$ is the ratio of the number of edges to the number of vertices. Myers and Thomason (2005), Norin, Reed, Thomason and Wood (2020), and Thomason and Wales (2019) determined the asymptotic behaviour of $c(H)$ for all polynomially dense graphs $H$, as well as almost all graphs $H$ of constant density. We explore the asymptotic behavior of the extremal function in the regime not covered by the above results, where in addition to having constant density the graph $H$ is in a graph class admitting strongly sublinear separators. We establish asymptotically tight bounds in many cases. For example, we prove that for every planar graph $H$, $$c(H) = (1+o(1))\cdot\max\left\{\frac{|V(H)|}{2},|V(H)| - α(H)\right\},$$ extending recent results of Haslegrave, Kim and Liu (2020). We also show that an asymptotically tight bound on the extremal function of graphs in minor-closed families proposed by Haslegrave, Kim and Liu (2020) is equivalent to a well studied open weakening of Hadwiger's conjecture.

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arXiv2022

Product structure of graph classes with bounded treewidth

We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the "underlying treewidth" of a graph class $\mathcal{G}$ to be the minimum non-negative integer $c$ such that, for some function $f$, for every graph ${G \in \mathcal{G}}$ there is a graph $H$ with ${\text{tw}(H) \leq c}$ such that $G$ is isomorphic to a subgraph of ${H \boxtimes K_{f(\text{tw}(G))}}$. We introduce disjointed coverings of graphs and show they determine the underlying treewidth of any graph class. Using this result, we prove that the class of planar graphs has underlying treewidth 3; the class of $K_{s,t}$-minor-free graphs has underlying treewidth $s$ (for ${t \geq \max\{s,3\}}$); and the class of $K_t$-minor-free graphs has underlying treewidth ${t-2}$. In general, we prove that a monotone class has bounded underlying treewidth if and only if it excludes some fixed topological minor. We also study the underlying treewidth of graph classes defined by an excluded subgraph or excluded induced subgraph. We show that the class of graphs with no $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a subdivided star, and that the class of graphs with no induced $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a star.

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arXiv2022

Bounds for the Twin-width of Graphs

Bonnet, Kim, Thomassé, and Watrigant (2020) introduced the twin-width of a graph. We show that the twin-width of an $n$-vertex graph is less than $(n+\sqrt{n\ln n}+\sqrt{n}+2\ln n)/2$, and the twin-width of an $m$-edge graph for a positive $m$ is less than $\sqrt{3m}+ m^{1/4} \sqrt{\ln m} / (4\cdot 3^{1/4}) + 3m^{1/4} / 2$. Conference graphs of order $n$ (when such graphs exist) have twin-width at least $(n-1)/2$, and we show that Paley graphs achieve this lower bound. We also show that the twin-width of the Erdős-Rényi random graph $G(n,p)$ with $1/n\leq p=p(n)\leq 1/2$ is larger than $2p(1-p)n - (2\sqrt{2}+\varepsilon)\sqrt{p(1-p)n\ln n}$ asymptotically almost surely for any positive $\varepsilon$. Lastly, we calculate the twin-width of random graphs $G(n,p)$ with $p\leq c/n$ for a constant $c<1$, determining the thresholds at which the twin-width jumps from $0$ to $1$ and from $1$ to $2$.

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arXiv2021

Counting cliques in $1$-planar graphs

The problem of maximising the number of cliques among $n$-vertex graphs from various graph classes has received considerable attention. We investigate this problem for the class of $1$-planar graphs where we determine precisely the maximum total number of cliques as well as the maximum number of cliques of any fixed size. We also precisely characterise the extremal graphs for these problems.

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arXiv2021

Obstructions for bounded branch-depth in matroids

DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a natural analogue of tree-depth of graphs. They conjectured that a matroid of sufficiently large branch-depth contains the uniform matroid $U_{n,2n}$ or the cycle matroid of a large fan graph as a minor. We prove that matroids with sufficiently large branch-depth either contain the cycle matroid of a large fan graph as a minor or have large branch-width. As a corollary, we prove their conjecture for matroids representable over a fixed finite field and quasi-graphic matroids, where the uniform matroid is not an option.

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