arXiv subjects

Wouter Cames van Batenburg

Publications and source records attributed to Wouter Cames van Batenburg.

At least 19 recordsLinked to original sources

arXiv2026

Counterexamples to the Albertson-Berman conjecture: minimum order, connectivity and an improved ratio bound

In 1979, Albertson and Berman conjectured that every planar graph $G$ contains an induced forest of order at least $|V(G)|/2$. This long-standing conjecture was recently disproved by several explicit counterexamples, which naturally led to several extremal and structural questions that we answer. We combine mathematical arguments and exhaustive computations to show that the minimum order of a counterexample is $29$. We also construct infinitely many $4$-connected $5$-edge-connected counterexamples (and show that the unique such counterexample of minimum order has order $41$), whereas previously all known counterexamples had vertex-connectivity at most $3$. Furthermore, we construct an infinite family of planar graphs on $n$ vertices whose maximum induced forests have order at most $\frac{25}{52}n$, thereby improving the previous best upper bound. This family also yields infinitely many counterexamples (for every integer $d \geq 7$) to a conjecture of Chappell and Pelsmajer concerning induced forests of maximum degree at most $d$.

math.CO↗

arXiv2026

Asymptotically attaining the Moore bound

For positive integers $d$ and $k$, let $n_k(d)$ be the maximum order of a graph of maximum degree at most $d$ and diameter at most $k$. We prove that $$ \lim_{d\to\infty}\frac{n_k(d)}{d^k}=1$$ for every fixed $k$, thereby resolving the asymptotic degree-diameter problem for fixed diameter and proving a conjecture of Bollob\'as. The lower bound comes from regular graphs $H_{k,q}$, indexed by prime powers $q$, whose vertices are partial flags in $\mathbb{F}_q^{\,2k+1}$. These graphs have diameter $k$ and order $|V(H_{k,q})| =(1+o(1))\Delta(H_{k,q})^k$. We also construct, for every fixed $\ell \ge 2$, graphs of maximum degree at most $d$ and line-graph diameter at most $\ell$ with $(1+o(1))d^{\ell}$ edges.

math.CO↗

arXiv2026

Domination-packing ratio for planar and unit disk graphs

The domination number $\gamma(G)$ of a graph $G$ is the smallest possible size of a vertex set that intersects every radius-$1$ ball of $G$, and the packing number $\rho(G)$ is the maximum number of pairwise vertex-disjoint radius-$1$ balls. We prove that $\frac{\gamma(G)}{\rho(G)}\le 5$ for every planar graph and $\frac{\gamma(G)}{\rho(G)} \le \frac{18\sqrt3}{\pi}\approx 9.924$ for every unit disk graph, thus yielding Erd\H{o}s-P\'osa-type bounds for the hypergraph of radius-$1$ balls in the two graph classes. This improves upon results of Guti\'errez and Paul, and D\'ucz and Gujgiczer, who in turn lowered bounds of Bonamy, Csik\'os, Gujgiczer and Yuditsky, and B\"ohme and Mohar. For both graph classes, the best known lower bound on the optimal constant remains $3$.

math.CO↗

arXiv2026

On the chromatic number of the union of comparability graphs

Resolving in a strong sense a problem of Gy\'arf\'as on the union of two perfect graphs, we prove that for every pair of positive integers $d$ and $k$, there is a graph $G$ with clique number $k$ and chromatic number $k^d$ that is the union of $d$ comparability graphs. We also show that the chromatic number can be replaced by the fractional chromatic number or $\frac{|V(G)|}{\alpha(G)}$.

math.CO↗

arXiv2026

Fractional list packing for layered graphs

The fractional list packing number $χ_{\ell}^{\bullet}(G)$ of a graph $G$ is a graph invariant that has recently arisen from the study of disjoint list-colourings. It measures how large the lists of a list-assignment $L:V(G)\rightarrow 2^{\mathbb{N}}$ need to be to ensure the existence of a `perfectly balanced' probability distribution on proper $L$-colourings, i.e., such that at every vertex $v$, every colour appears with equal probability $1/|L(v)|$. In this work we give various bounds on $χ_{\ell}^{\bullet}(G)$, which admit strengthenings for correspondence and local-degree versions. As a corollary, we improve theorems on the related notion of flexible list colouring. In particular we study Cartesian products and $d$-degenerate graphs, and we prove that $χ_{\ell}^{\bullet}(G)$ is bounded from above by the pathwidth of $G$ plus one. The correspondence analogue of the latter is false for treewidth instead of pathwidth.

math.CO↗

arXiv2026

Hat guessing with proper colorings

We initiate the study of the hat guessing number of a graph where the adversary is only allowed to provide a proper coloring of the graph. This is the largest number $q$ for which there is a guessing strategy on each vertex that only depends on its neighborhood, such that for every proper coloring of the graph with $q$ colors at least one vertex guesses its color correctly. In this variation, we prove that the hat guessing number of the complete graphs on $n$ vertices is $2n - 1$, which is roughly twice the classical hat guessing number of the complete graph. Our winning strategy is related to finding perfect matchings between the middle layers of the boolean poset of dimension $2n - 1$. We prove that the hat guessing number of all trees on $n \geq 3$ vertices is equal to $4$. We derive general upper bounds in terms of the number of vertices, chromatic number, and maximum degree, and obtain improved bounds for book graphs. Using our results and an ILP formulation of the problem, we determine the exact hat guessing number for all graphs on at most $4$ vertices, give bounds on graphs on $5$ vertices, and suggest some open problems.

math.CO↗

arXiv2026

Disjoint Correspondence Colorings for $K_5$-Minor-free Graphs

Thomassen famously proved that every planar graph is 5-choosable. We explore variants of this result, focusing on finding disjoint correspondence colorings, in the more general class of $K_5$-minor-free graphs. Correspondence colorings generalize list colorings as follows. Given a graph $G$ and a positive integer $t$, a correspondence $t$-cover $\textbf{M}$ assigns to each $v\in V(G)$ a set of allowable colors $\{1_v,\ldots,t_v\}$ and to each edge $vw\in E(G)$ a matching between $\{1_v,\ldots,t_v\}$ and $\{1_w,\ldots,t_w\}$. An $\textbf{M}$-coloring $φ$ picks for each vertex $v$ a color $φ(v)$ (from the set $\{1_v,\ldots,t_v\}$) such that for each edge $vw\in E(G)$ the colors $φ(v),φ(w)$ are not matched to each other. Two $\textbf{M}$-colorings $φ_1,φ_2$ of $G$ are called disjoint if $φ_1(v)\neφ_2(v)$ for all $v\in V(G)$. For every $K_5$-minor-free graph $G$ and every correspondence 6-cover $\textbf{M}$ of $G$, we construct 3 pairwise disjoint $\textbf{M}$-colorings $φ_1,φ_2,φ_3$. In contrast, we provide examples of $K_5$-minor-free graphs and correspondence 5-covers $\textbf{M}$ that do not admit 3 disjoint $\textbf{M}$-colorings.

math.CO↗

arXiv2025

The chromatic number of finite projective spaces

The chromatic number of the finite projective space $\mathrm{PG}(n-1,q)$, denoted $\chi_q(n)$, is the minimum number of colors needed to color its points so that no line is monochromatic. We prove subadditivity of $\chi_q(n)$ with respect to $n$, and then establish the following stronger recursive bound: \[ \chi_q(n)\le \chi_q(d)+\chi_q(n+1-d)-1 \] for all $1 \leq d < n$. We use it to prove new upper bounds on $\chi_q(n)$. For $q = 2$, using this recursion we prove that \[ \chi_2(n) \le \lfloor 2n/3 \rfloor + 1 \] for all $n \ge 2$, and we show that this bound is tight for all $n \le 7$. In particular, our result recovers all previously known cases for $n \le 6$ and resolves the first open case $n = 7$. It also disproves a conjecture of Haddad that $\chi_2(n) = n - 1$ for all $n \geq 4$, in a strong sense. On the lower-bound side, using a connection with multicolor Ramsey numbers for triangles, we note that \[ \chi_2(n) \ge (1 - o(1))\,\frac{n}{\log n}.\] We also consider $\chi_q(t;n)$, the minimum number of colors needed to color the points of $\mathrm{PG}(n-1,q)$ with no monochromatic $(t - 1)$-dimensional subspace, and establish an equivalence between $\chi_q(t;n)$ and the multicolor vector-space Ramsey numbers $R_q(t;k)$. Using this equivalence together with new upper bounds on $\chi_q(t;n)$, we improve, for every fixed $t$ and $q$, the best known lower bounds on $R_q(t;k)$ from $\Omega_{q,t}(\log k)$ to $\Omega(k)$.

math.CO↗

arXiv2025

An improved upper bound for the multicolour Ramsey number of odd cycles

We show that the $k$-colour Ramsey number of an odd cycle of length $2 \ell + 1$ is at most $(4 \ell)^k \cdot k^{k/\ell}$. This proves a conjecture of Fox and is the first improvement in the exponent that goes beyond an absolute constant factor since the work of Bondy and Erdős from 1973.

math.CO↗

arXiv2025

Shallow brambles

A graph class $\mathcal{C}$ has polynomial expansion if there is a polynomial function $f$ such that for every graph $G\in \mathcal{C}$, each of the depth-$r$ minors of $G$ has average degree at most $f(r)$. In this note, we study bounded-radius variants of some classical graph parameters such as bramble number, linkedness and well-linkedness, and we show that they are pairwise polynomially related. Furthermore, in a monotone graph class with polynomial expansion they are all uniformly bounded by a polynomial in $r$.

math.CO↗

arXiv2025

Reconfiguration of List Colourings

Given a proper (list) colouring of a graph $G$, a recolouring step changes the colour at a single vertex to another colour (in its list) that is currently unused on its neighbours, hence maintaining a proper colouring. Suppose that each vertex $v$ has its own private list $L(v)$ of allowed colours such that $|L(v)|\ge \mbox{deg}(v)+1$. We prove that if $G$ is connected and its maximum degree $Δ$ is at least $3$, then for any two proper $L$-colourings in which at least one vertex can be recoloured, one can be transformed to the other by a sequence of $O(|V(G)|^2)$ recolouring steps. We also show that reducing the list-size of a single vertex $w$ to $\mbox{deg}(w)$ can lead to situations where the space of proper $L$-colourings is `shattered'. Our results can be interpreted as showing a sharp phase transition in the Glauber dynamics of proper $L$-colourings of graphs. This constitutes a `local' strengthening and generalisation of a result of Feghali, Johnson, and Paulusma, which considered the situation where the lists are all identical to $\{1,\ldots,Δ+1\}$.

math.CO↗

arXiv2024

Sharp Bounds on Lengths of Linear Recolouring Sequences

A recolouring sequence, between $k$-colourings $α$ and $β$ of a graph $G$, transforms $α$ into $β$ by recolouring one vertex at a time, such that after each recolouring step we again have a proper $k$-colouring of $G$. The diameter of the $k$-recolouring graph, $\textrm{diam}~\mathcal{C}_k(G)$, is the maximum over all pairs $α$ and $β$ of the minimum length of a recolouring sequence from $α$ to $β$. Much previous work has focused on determining the asymptotics of $\textrm{diam}~\mathcal{C}_k(G)$: Is it $Θ(|G|)$? Is it $Θ(|G|^2)$? Or even larger? Here we focus on graphs for which $\textrm{diam}~\mathcal{C}_k(G)=Θ(|G|)$, and seek to determine more precisely the multiplicative constant implicit in the $Θ()$. In particular, for each $k\ge 3$, for all positive integers $p$ and $q$ we exactly determine $\textrm{diam}~\mathcal{C}_k(K_{p,q})$, up to a small additive constant. We also sharpen a recolouring lemma that has been used in multiple papers, proving an optimal version. This improves the multiplicative constant in various prior results. Finally, we investigate plausible relationships between similar reconfiguration graphs.

math.CO↗

arXiv2023

Disjoint list-colorings for planar graphs

One of Thomassen's classical results is that every planar graph of girth at least $5$ is 3-choosable. One can wonder if for a planar graph $G$ of girth sufficiently large and a $3$-list-assignment $L$, one can do even better. Can one find $3$ disjoint $L$-colorings (a packing), or $2$ disjoint $L$-colorings, or a collection of $L$-colorings that to every vertex assigns every color on average in one third of the cases (a fractional packing)? We prove that the packing is impossible, but two disjoint $L$-colorings are guaranteed if the girth is at least $8$, and a fractional packing exists when the girth is at least $6.$ For a graph $G$, the least $k$ such that there are always $k$ disjoint proper list-colorings whenever we have lists all of size $k$ associated to the vertices is called the list packing number of $G$. We lower the two-times-degeneracy upper bound for the list packing number of planar graphs of girth $3,4$ or $5$. As immediate corollaries, we improve bounds for $ε$-flexibility of classes of planar graphs with a given girth. For instance, where previously Dvořák et al. proved that planar graphs of girth $6$ are (weighted) $ε$-flexibly $3$-choosable for an extremely small value of $ε$, we obtain the optimal value $ε=\frac{1}{3}$. Finally, we completely determine and show interesting behavior on the packing numbers for $H$-minor-free graphs for some small graphs $H.$

math.CO↗

arXiv2023

Optimally Reconfiguring List and Correspondence Colourings

The reconfiguration graph $\mathcal{C}_k(G)$ for the $k$-colourings of a graph $G$ has a vertex for each proper $k$-colouring of $G$, and two vertices of $\mathcal{C}_k(G)$ are adjacent precisely when those $k$-colourings differ on a single vertex of $G$. Much work has focused on bounding the maximum value of ${\rm{diam}}~\mathcal{C}_k(G)$ over all $n$-vertex graphs $G$. We consider the analogous problems for list colourings and for correspondence colourings. We conjecture that if $L$ is a list-assignment for a graph $G$ with $|L(v)|\ge d(v)+2$ for all $v\in V(G)$, then ${\rm{diam}}~\mathcal{C}_L(G)\le n(G)+μ(G)$. We also conjecture that if $(L,H)$ is a correspondence cover for a graph $G$ with $|L(v)|\ge d(v)+2$ for all $v\in V(G)$, then ${\rm{diam}}~\mathcal{C}_{(L,H)}(G)\le n(G)+τ(G)$. (Here $μ(G)$ and $τ(G)$ denote the matching number and vertex cover number of $G$.) For every graph $G$, we give constructions showing that both conjectures are best possible. Our first main result proves the upper bounds (for the list and correspondence versions, respectively) ${\rm{diam}}~\mathcal{C}_L(G)\le n(G)+2μ(G)$ and ${\rm{diam}}~\mathcal{C}_{(L,H)}(G)\le n(G)+2τ(G)$. Our second main result proves that both conjectured bounds hold, whenever all $v$ satisfy $|L(v)|\ge 2d(v)+1$. We conclude by proving one or both conjectures for various classes of graphs such as complete bipartite graphs, subcubic graphs, cactuses, and graphs with bounded maximum average degree.

math.CO↗

arXiv2023

Packing list-colourings

List colouring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list-colouring, we seek many in parallel. Our explorations have uncovered a potentially rich seam of interesting problems spanning chromatic graph theory. Given a $k$-list-assignment $L$ of a graph $G$, which is the assignment of a list $L(v)$ of $k$ colours to each vertex $v\in V(G)$, we study the existence of $k$ pairwise-disjoint proper colourings of $G$ using colours from these lists. We may refer to this as a \emph{list-packing}. Using a mix of combinatorial and probabilistic methods, we set out some basic upper bounds on the smallest $k$ for which such a list-packing is always guaranteed, in terms of the number of vertices, the degeneracy, the maximum degree, or the (list) chromatic number of $G$. (The reader might already find it interesting that such a minimal $k$ is well defined.) We also pursue a more focused study of the case when $G$ is a bipartite graph. Our results do not yet rule out the tantalising prospect that the minimal $k$ above is not too much larger than the list chromatic number. Our study has taken inspiration from study of the strong chromatic number, and we also explore generalisations of the problem above in the same spirit.

math.CO↗

arXiv2023

List packing number of bounded degree graphs

We investigate the list packing number of a graph, the least $k$ such that there are always $k$ disjoint proper list-colourings whenever we have lists all of size $k$ associated to the vertices. We are curious how the behaviour of the list packing number contrasts with that of the list chromatic number, particularly in the context of bounded degree graphs. The main question we pursue is whether every graph with maximum degree $Δ$ has list packing number at most $Δ+1$. Our results highlight the subtleties of list packing and the barriers to, for example, pursuing a Brooks'-type theorem for the list packing number.

math.CO↗

arXiv2021

Maximising line subgraphs of diameter at most $t$

We wish to bring attention to a natural but slightly hidden problem, posed by Erdős and Nešetřil in the late 1980s, an edge version of the degree--diameter problem. Our main result is that, for any graph of maximum degree $Δ$ with more than $1.5 Δ^t$ edges, its line graph must have diameter larger than $t$. In the case where the graph contains no cycle of length $2t+1$, we can improve the bound on the number of edges to one that is exact for $t\in\{1,2,3,4,6\}$. In the case $Δ=3$ and $t=3$, we obtain an exact bound. Our results also have implications for the related problem of bounding the distance-$t$ chromatic index, $t>2$; in particular, for this we obtain an upper bound of $1.941Δ^t$ for graphs of large enough maximum degree $Δ$, markedly improving upon earlier bounds for this parameter.

math.CO↗

arXiv2021

Strong chromatic index and Hadwiger number

We investigate the effect of a fixed forbidden clique minor upon the strong chromatic index, both in multigraphs and in simple graphs. We conjecture for each $k\ge 4$ that any $K_k$-minor-free multigraph of maximum degree $Δ$ has strong chromatic index at most $\frac32(k-2)Δ$. We present a construction certifying that if true the conjecture is asymptotically sharp as $Δ\to\infty$. In support of the conjecture, we show it in the case $k=4$ and prove the statement for strong clique number in place of strong chromatic index. By contrast, we make a basic observation that for $K_k$-minor-free simple graphs, the problem of strong edge-colouring is "between" Hadwiger's Conjecture and its fractional relaxation. For $k\geq5$, we also show that $K_k$-minor-free multigraphs of edge-diameter at most $2$ have strong clique number at most $(k-\frac{1}{2})Δ$.

math.CO↗