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David R. Wood

Publications and source records attributed to David R. Wood.

At least 19 recordsLinked to original sources
arXiv2026

Proof of the Clustered Hadwiger Conjecture

Hadwiger's Conjecture asserts that every $K_h$-minor-free graph is properly $(h-1)$-colourable. We prove the following improper analogue of Hadwiger's Conjecture: for fixed $h$, every $K_h$-minor-free graph is $(h-1)$-colourable with monochromatic components of bounded size. The number of colours is best possible regardless of the size of monochromatic components. It solves an open problem of Edwards, Kang, Kim, Oum and Seymour [SIAM J. Disc. Math. 2015], and concludes a line of research initiated in 2007. Similarly, for fixed $t\geq s$, we show that every $K_{s,t}$-minor-free graph is $(s+1)$-colourable with monochromatic components of bounded size. The number of colours is best possible, solving an open problem of van de Heuvel and Wood [J. London Math. Soc. 2018]. We actually prove a single theorem from which both of the above results are immediate corollaries. For an excluded apex minor, we strengthen the result as follows: for fixed $t\geq s\geq 3$, and for any fixed apex graph $X$, every $K_{s,t}$-subgraph-free $X$-minor-free graph is $(s+1)$-colourable with monochromatic components of bounded size. The number of colours is again best possible.

math.CO↗
arXiv2026

The Erdős--Sós Theorem

We present an exposition of a proof, discovered by GPT-6 Astra, of the Erdős--Sós Conjecture, which states that every graph with average degree greater than $t-2$ contains every tree on $t\geq 2$ vertices.

math.HO↗
arXiv2026

Countable Graphs with Finite Path-width: Characterisation and Universality

We study path-width and the closely related parameter line-width in countably infinite graphs. Our first result characterises the graphs of finite path-width: they are the graphs that do not have infinitely many vertices of infinite degree, do not have infinitely many pairwise disjoint infinite paths, and contain no subdivision of some finite tree of maximum degree 3. We then investigate universality under the subgraph relation for graphs of bounded path-width or line-width. In particular, we prove that there exists a universal graph with line-width $\mathcal{O}(k^2)$ for the class of graphs with line-width at most $k$. In contrast, we show that no graph of finite path-width is universal for the class of locally finite graphs with path-width $1$. Finally, we show that for each $k\geq 2$, every universal graph for the class of graphs with path-width at most $k$ has line-width at least $k + 1$.

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arXiv2026

Optimal tree-decompositions with bags of bounded pathwidth

We show that every planar graph has a tree-decomposition with optimal width such that the subgraph induced by each bag has pathwidth at most 3. This bound is best possible, and for tree-decompositions that satisfy a certain minimality condition, we in fact give a precise description of the possible structures in each bag. Moreover, we show that the union of any $k$ bags has pathwidth $O(k)$. We also show that graphs excluding a fixed double-apex-forest minor have a tree-decomposition with optimal width such that the subgraph induced by each bag has bounded pathwidth. This includes graphs embeddable on any fixed surface. As a byproduct of our machinery, we give a new proof of the linear grid minor theorem for planar graphs.

math.CO↗
arXiv2026

3-Colouring Planar Graphs

We show that every $n$-vertex planar graph is 3-colourable with monochromatic components of size $O(n^{4/9})$. The best previous bound was $O(n^{1/2})$ due to Linial, Matoušek, Sheffet and Tardos [Combin. Probab. Comput., 2008].

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arXiv2026

Adjacency labelling for proper minor-closed graph classes

We show that every proper minor-closed class of graphs admits a $(1+o(1))\log_2 n$-bit adjacency labelling scheme. Equivalently, for every proper minor-closed class $\mathcal{G}$ and every positive integer $n$ there exists an $n^{1+o(1)}$-vertex graph $U$ such that every $n$-vertex graph in $\mathcal{G}$ is isomorphic to an induced subgraph of $U$. Both results are optimal up to the lower order term. They generalize the corresponding results for planar graphs and apex-minor-free classes (Dujmović et al., J.~ACM 2021) to all proper minor-closed classes, answering the open question raised in that paper and anticipated earlier by Bonamy, Gavoille, and Pilipczuk (SODA 2020).

cs.DM↗
arXiv2026

Universality in minor-closed graph classes

Stanislaw Ulam asked whether there exists a universal countable planar graph (that is, a countable planar graph that contains every countable planar graph as a subgraph). János Pach (1981) answered this question in the negative. We strengthen this result by showing that every countable graph that contains all countable planar graphs must contain (i) an infinite complete graph as a minor, and (ii) a subdivision of the complete graph $K_t$ with multiplicity $t$, for every finite $t$. On the other hand, we construct a countable graph that contains all countable planar graphs and has several key properties such as linear colouring numbers, linear expansion, and every finite $n$-vertex subgraph has a balanced separator of size $O(\sqrt{n})$. The graph is $T_6\boxtimes P_{\!\infty}$, where $T_k$ is the universal treewidth-$k$ countable graph (which we define explicitly), $P_{\!\infty}$ is the 1-way infinite path, and $\boxtimes$ denotes the strong product. More generally, for every positive integer $t$ we construct a countable graph that contains every countable $K_t$-minor-free graph and has the above key properties. Our final contribution is a construction of a countable graph that contains every countable $K_t$-minor-free graph as an induced subgraph, has linear colouring numbers and linear expansion, and contains no subdivision of the countably infinite complete graph (implying (ii) above is best possible).

math.CO↗
arXiv2026

Clustered Graph Coloring and Layered Treewidth

A graph coloring has bounded clustering if each monochromatic component has bounded size. This paper studies such a coloring, where the number of colors depends on an excluded complete bipartite subgraph. This is a much weaker assumption than previous works, where typically the number of colors depends on an excluded minor. This paper focuses on graph classes with bounded layered treewidth, which include planar graphs, graphs of bounded Euler genus, graphs embeddable on a fixed surface with a bounded number of crossings per edge, amongst other examples. Our main theorem says that for fixed integers $s,t,k$, every graph with layered treewidth at most $k$ and with no $K_{s,t}$ subgraph is $(s+2)$-colorable with bounded clustering. The $s=3$ case implies that every graph with a drawing on a fixed surface with a bounded number of crossings per edge is 5-colorable with bounded clustering. Our main theorem is also a critical component in two companion papers that study clustered coloring of graphs with no $K_{s,t}$ subgraph and excluding a fixed minor, odd minor or topological minor.

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arXiv2026

Non-Homotopic Drawings of Multigraphs

A multigraph drawn in the plane is non-homotopic if no two edges connecting the same pair of vertices can be continuously deformed into each other without passing through a vertex, and is $k$-crossing if every pair of edges (self-)intersects at most $k$ times. We prove that the number of edges in an $n$-vertex non-homotopic $k$-crossing multigraph is at most $6^{13 n (k + 1)}$, which is a substantial improvement over previous upper bounds. We also study this problem in the setting of monotone drawings where every edge is an x-monotone curve. We show that the number of edges, $m$, in such a drawing is at most $2 \binom{2n}{k + 1}$ and the number of crossings is $Ω\bigl(\frac{m^{2 + 1/k}}{n^{1 + 1/k}}\bigr)$. For fixed $k$ these bounds are both best possible up to a constant multiplicative factor.

math.CO↗
arXiv2026

The grid-minor theorem revisited

We prove that for every planar graph $X$ of treedepth $h$, there exists a positive integer $c$ such that for every $X$-minor-free graph $G$, there exists a graph $H$ of treewidth at most $f(h)$ such that $G$ is isomorphic to a subgraph of $H\boxtimes K_c$. This is a qualitative strengthening of the Grid-Minor Theorem of Robertson and Seymour (JCTB 1986), and treedepth is the optimal parameter in such a result. As an example application, we use this result to improve the upper bound for weak coloring numbers of graphs excluding a fixed graph as a minor.

math.CO↗
arXiv2026

Tree-partitions of graphs with given pathwidth

Graphs with bounded treewidth and bounded maximum degree are known to have tree-partitions of bounded width. What can be said if the bounded treewidth assumption is strengthened to bounded pathwidth? We prove that every graph with bounded pathwidth and bounded maximum degree has a tree-partition of bounded width, with the extra property that the underlying tree has bounded pathwidth. Moreover, we prove a lower bound showing that the bound on the pathwidth of the underlying tree is within a constant factor of optimal.

math.CO↗
arXiv2026

The Dominating 4-Colour Theorem

A "dominating $K_t$-model" in a graph $G$ is a sequence $(T_1,\dots,T_t)$ of pairwise vertex-disjoint connected subgraphs of $G$, such that whenever $1\leq i<j\leq t$ every vertex in $T_j$ has a neighbour in $T_i$. Replacing "every vertex in $T_j$" by "some vertex in $T_j$" retrieves the standard definition of $K_t$-model, which is equivalent to a $K_t$-minor in $G$. We prove that every graph with no dominating $K_5$-model is $4$-colourable. This generalises and is significantly stronger than the 4-colour theorem for planar graphs or for graphs with no $K_5$-minor. It also makes progress towards Hajós' conjecture on $K_5$-subdivisions in $5$-chromatic graphs.

math.CO↗
arXiv2026

Tree decompositions with small width, spread, order and degree

Tree-decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. The main property of tree-decompositions is the width (the maximum size of a bag minus 1). We show that every graph has a tree-decomposition with near-optimal width, where each vertex appears in few bags. In particular, every graph with treewidth $k$ has a tree-decomposition with width at most $14k+13$, where each vertex $v$ appears in at most $\text{deg}(v)+1$ bags. This improves an exponential bound by Ding and Oporowski [1995] to linear, and establishes a conjecture of theirs in a strong sense. In a second result, we show that every graph with treewidth $k$ has a tree-decomposition with width at most $3k-1$, where on average each vertex appears in at most three bags.

math.CO↗
arXiv2026

Tree-partitions and small-spread tree-decompositions

Tree-decompositions and treewidth are of fundamental importance in structural and algorithmic graph theory. The "spread" of a tree-decomposition is the minimum integer $s$ such that every vertex lies in at most $s$ bags. A tree-decomposition is "domino" if it has spread 2, which is the smallest interesting value of spread. So that spread 1 becomes interesting, one can relax the definition of tree-decomposition to "tree-partition", which allows the endpoints of each edge to be in the same bag or adjacent bags, while demanding that each vertex appears in exactly one bag. Ding and Oporowski [1995] showed that every graph $G$ with treewidth $k$ and maximum degree $Δ$ has a tree-partition with width $O(kΔ)$. We prove the same result with an improved constant, and with the extra property that the underlying tree has maximum degree $O(Δ)$ and $O(|V(G)|/kΔ)$ vertices. This result implies (with an improved constant) the best known upper bound on the domino treewidth of $O(kΔ^2)$, due to Bodlaender [1999]. Moreover, solving an open problem of Bodlaender, we show this upper bound is best possible, by exhibiting graphs with domino treewidth $Ω(kΔ^2)$ for $k\geqslant 2$. On the other hand, allowing the spread to be a function of $k$, we show that width $O(kΔ)$ can be achieved. This result exploits a connection to chordal completions, which we show is best possible, a result of independent interest.

math.CO↗
arXiv2026

Planar graphs in blowups of fans

We show that every $n$-vertex planar graph is contained in the graph obtained from a fan by blowing up each vertex by a complete graph of order $O(\sqrt{n}\log^2 n)$. Equivalently, every $n$-vertex planar graph $G$ has a set $X$ of $O(\sqrt{n}\log^2 n)$ vertices such that $G-X$ has bandwidth $O(\sqrt{n}\log^2 n)$. We in fact prove the same result for any proper minor-closed class, and we prove more general results that explore the trade-off between $X$ and the bandwidth of $G-X$. The proofs use three key ingredients. The first is a new local sparsification lemma, which shows that every $n$-vertex planar graph $G$ has a set of $O((n\log n)/δ)$ vertices whose removal results in a graph with local density at most $δ$. The second is a generalization of a method of Feige and Rao that relates bandwidth and local density using volume-preserving Euclidean embeddings. The third ingredient is graph products, which are a key tool in the extension to any proper minor-closed class.

math.CO↗
arXiv2026

On Universal Graphs for Trees and Tree-Like Graphs

Chung and Graham [J. London Math. Soc. 1983] claimed to prove that there exists an $n$-vertex graph $G$ with $ \frac{5}{2}n \log_2 n + O(n)$ edges that contains every $n$-vertex tree as a subgraph. Frati, Hoffmann and Tóth [Combin. Probab. Comput. 2023] discovered an error in the proof. By adding more edges to $G$ the error can be corrected, bringing the number of edges in $G$ to $\frac{7}{2}n \log_2 n + O(n). $ We make the first improvement to Chung and Graham's bound in over four decades by showing that there exists an $n$-vertex graph with $ \frac{14}{5}n \log_2 n + O(n) $ edges that contains every $n$-vertex tree as a subgraph. Furthermore, we generalise this bound for treewidth-$k$ graphs by showing that there exists a graph with $O(kn\log(n/k+1))$ edges that contains every $n$-vertex treewidth-$k$ graph as a subgraph. This is best possible in the sense that $Ω(kn\log(n/k+1))$ edges are required.

math.CO↗
arXiv2025

Verifying Hadwiger's Conjecture for Examples of Graphs with $α(G) = 2$

Hadwiger's Conjecture states that every graph with chromatic number $k$ contains a complete graph on $k$ vertices as a minor. This conjecture is a tremendous strengthening of the Four-Colour Theorem and is regarded as one of the most important open problems in graph theory. The case of Hadwiger's Conjecture for graphs with $α(G) = 2$ has garnered much attention. Seymour writes: ``My own belief is, if Hadwiger's Conjecture is true for graphs with stability number two then it is probably true in general, so it would be very nice to decide this case.'' This paper presents several tools useful for proving that a graph $G$ with $α(G) = 2$ satisfies Hadwiger's Conjecture. In doing so, we survey and generalise several classical results on the $α(G) = 2$ case of Hadwiger's Conjecture. Further, we apply these tools to prove variants of Hadwiger's Conjecture for several noteworthy classes of graphs with $α(G) = 2$. In particular, we prove Hadwiger's Conjecture for inflations of the complements of the following graphs: graphs with girth at least $5$, triangle-free Kneser graphs, and the Clebsch, Mesner, and Gewirtz graphs. This paper also highlights classes of graphs with $α(G) = 2$ where it is unknown if Hadwiger's Conjecture holds.

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