We show that, for every fixed graph H, every n-vertex graph G that excludes H as a minor is 3-colourable with clustering O_H(n^4/9). That is, there exists a function f such that for every graph H, every n≥ 1, every n-vertex graph G that excludes H as a minor has a vertex colouring with 3 colours in which each monochromatic component has size at most f(H)· n^4/9. This generalizes a recent result of Dujmović, Morin, Norin, and Wood (arXiv:2507.03163) from planar graphs to all proper minor-closed graph classes and is the first improvement on clustered 3-colouring of proper minor-closed graph classes since the upper bound of O_H(√(n)) due to Linial, Matoušek, Sheffet, and Tardos (Comb. Prob. Comput., 17(4):577–589, 2008).

We prove that there exist functions f:ℕ^2→ℕ and g:ℕ→ℕ such that for all positive integers k, d, and ℓ≥3, every graph G either contains k cycles of length at least ℓ that are pairwise at distance greater than d, or admits a subset of vertices X with |X|≤ f(k,ℓ) such that G-B_G(X,g(d)) contains no cycle of length at least ℓ, where B_G(X,r) denotes the ball of radius r around X. This generalizes a theorem of Dujmović, Joret, Micek, and Morin (2024), which established the ℓ=3 case. Moreover, we prove that the theorem holds with f(k,ℓ)∈𝒪(ℓ klog k) and g(d)∈𝒪(d). The linear bound on g is best possible, while the bound on f is optimal as a function of k for every fixed ℓ. In particular, for ℓ=3 our result improves the previous bound of 𝒪(k^18𝗉𝗈𝗅𝗒𝗅𝗈𝗀 k) by Dujmović et al.

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 𝒢 and every positive integer n there exists an n^1+o(1)-vertex graph U such that every n-vertex graph in 𝒢 is isomorphic to an induced subgraph of U. Both results are optimal up to the lower order term.

A centred colouring of a graph is a vertex colouring in which every connected subgraph contains a vertex whose colour is unique and a linear colouring is a vertex colouring in which every (not-necessarily induced) path contains a vertex whose colour is unique. For a graph , the centred chromatic number and the linear chromatic number denote the minimum number of distinct colours required for a centred, respectively, linear colouring of . From these definitions, it follows immediately that for every graph . The centred chromatic number is equivalent to treedepth and has been studied extensively. Much less is known about linear colouring. Kun and colleagues prove that for any graph and conjecture that . Their upper bound was subsequently improved by Czerwi & nacute;ski and colleagues to . The proof of both upper bounds relies on establishing a lower bound on the linear chromatic number of pseudogrids, which appear in the proof due to their critical relationship to treewidth. Specifically, Kun and colleagues prove that pseudogrids have linear chromatic number . Our main contribution is establishing a tight bound on the linear chromatic number of pseudogrids, specifically for every pseudogrid . As a consequence we improve the general bound for all graphs to . In addition, this tight bound gives further evidence in support of Kun and colleagues' conjecture (above) that the centred chromatic number (i.e., the treedepth) of any graph is upper bounded by a linear function of its linear chromatic number.

We show that every $n$-vertex triangulation has a connected dominating set of size at most $10n/21$. Equivalently, every $n$ vertex triangulation has a spanning tree with at least $11n/21$ leaves. Prior to the current work, the best known bounds were $n/2$, which follows from work of Albertson, Berman, Hutchinson, and Thomassen (J. Graph Theory \textbf{14}(2):247--258). One immediate consequence of this result is an improved bound for the SEFENOMAP graph drawing problem of Angelini, Evans, Frati, and Gudmundsson (J. Graph Theory \textbf{82}(1):45--64). As a second application, we show that for every set $P$ of $\lceil 11n/21\rceil$ points in $\R^2$ every $n$-vertex planar graph has a one-bend non-crossing drawing in which some set of $11n/21$ vertices is drawn on the points of $P$. The main result extends to $n$-vertex triangulations of genus-$g$ surfaces, and implies that these have connected dominating sets of size at most $10n/21+O(\sqrt{gn})$.

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. We give three applications of this result: (1) improved upper bounds for the weak coloring numbers of graphs excluding a given minor, (2) an improved product structure theorem for apex-minor-free graphs, and (3) improved upper bounds for the p -centered chromatic number of graphs excluding a given minor.

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(√(n log² n)). Equivalently, every n-vertex planar graph G has a set X of O(√(n log² n)) vertices such that G−X has bandwidth O(√(n log² 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.

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].

Motivated by recent developments regarding the product structure of planar graphs, we study relationships between treewidth, grid minors, and graph products. We show that the Cartesian product of any two connected $n$-vertex graphs contains an $\Omega(\sqrt{n})\times\Omega(\sqrt{n})$ grid minor. This result is tight: The lexicographic product (which includes the Cartesian product as a subgraph) of a star and any $n$-vertex tree has no $\omega(\sqrt{n})\times\omega(\sqrt{n})$ grid minor.

The workshop provided a venue for discussing several recent major developments in graph theory, with a primary focus on results concerning the notion of twin-width and hereditary properties of graphs. Both areas have seen a very rapid development recently, as reflected in the many results presented during the workshop. In addition to many interesting talks spanning the whole breadth of graph theory, the workshop also offered valuable collaboration opportunities, which also engaged early career researchers.

We study the impact of forbidding short cycles to the edge density of k-planar graphs; a k-planar graph is one that can be drawn in the plane with at most k crossings per edge. Specifically, we consider three settings, according to which the forbidden substructures are 3-cycles, 4-cycles or both of them (i.e., girth ≥ 5). For all three settings and all k ∈ {1,2,3}, we present lower and upper bounds on the maximum number of edges in any k-planar graph on n vertices. Our bounds are of the form c\sqrt{k}n, for some explicit constant c that depends on k and on the setting. For general k ≥ 4 our bounds are of the form c\sqrt{k}n, for some explicit constant c. These results are obtained by leveraging different techniques, such as the discharging method, the recently introduced density formula for non-planar graphs, and new upper bounds for the crossing number of 2-- and 3-planar graphs in combination with corresponding lower bounds based on the Crossing Lemma.

A subset $S$ of vertices in a planar graph $G$ is a free set if, for every set $P$ of $|S|$ points in the plane, there exists a straight-line crossing-free drawing of $G$ in which vertices of $S$ are mapped to distinct points in $P$. In this survey, we review - several equivalent definitions of free sets, - results on the existence of large free sets in planar graphs and subclasses of planar graphs, - and applications of free sets in graph drawing. The survey concludes with a list of open problems in this still very active research area.

The rectilinear crossing number of G is the minimum number of crossings in a straight-line drawing of G. A single-crossing graph is a graph whose crossing number is at most one. We prove that every n-vertex graph G that excludes a single-crossing graph as a minor has rectilinear crossing number O(Delta n), where Delta is the maximum degree of G. This dependence on n and. is best possible. The result applies, for example, to K-5-minor-free graphs, and bounded treewidth graphs. Prior to our work, the only bounded degree minor-closed families known to have linear rectilinear crossing number were bounded degree graphs of bounded treewidth as well as bounded degree K3,3-minor-free graphs. In the case of bounded treewidth graphs, our O(Delta n) result is again tight and it improves on the previous best known bound of O(Delta(2)n) by Wood and Telle, 2007.

The study of nonplanar drawings of graphs with restricted crossing configurations is a well-established topic in graph drawing, often referred to as beyond-planar graph drawing. One of the most studied types of drawings in this area are the $k$-planar drawings $(k \geq 1)$, where each edge cannot cross more than $k$ times. We generalize $k$-planar drawings, by introducing the new family of min-$k$-planar drawings. In a min-$k$-planar drawing edges can cross an arbitrary number of times, but for any two crossing edges, one of the two must have no more than $k$ crossings. We prove a general upper bound on the number of edges of min-$k$-planar drawings, a finer upper bound for $k=3$, and tight upper bounds for $k=1,2$. Also, we study the inclusion relations between min-$k$-planar graphs (i.e., graphs admitting min-$k$-planar drawings) and $k$-planar graphs.In our setting, we only allow simple drawings, that is, any two edges cross at most once, no two adjacent edges cross, and no three edges intersect at a common point.

Graph drawing beyond planarity is a research area that has received an increasing attention in the last twenty years, driven by the necessity to mitigate the visual complexity inherent in geometric representations of non-planar graphs. This research area stems from the study of graph layouts with forbidden crossing configurations, a well-established subject in geometric and topological graph theory. In this context, the contribution of this paper is as follows: 1) We introduce a new hierarchy of graph families, called k(+) -real face graphs; for any integer k >= 1 , a graph G is a k(+) -real face graph if it admits a drawing Gamma in the plane such that the boundary of each face (formed by vertices, crossings, and edges) contains at least k vertices of G (" k(+) " stands for k or more); 2) We give tight upper bounds on the edge density of k(+) -real face graphs, namely we prove that n-vertex 1+ -real face and 2+ -real face graphs have at most 5n-10 and 4n-8 edges, respectively. Furthermore, in a constrained scenario in which all vertices must lie on the boundary of the external face, 1+ -real face and 2+ -real face graphs have at most 3n-6 and 2.5n-4 edges, respectively; 3) We characterize the complete graphs that admit a k(+) -real face drawing or an outer k(+) -real face drawing for any k >= 1 . We also provide a clear picture for the majority of complete bipartite graphs; and 4) We establish relationships between k(+) -real face graphs and other prominent beyond-planar graph families; notably, we show that for any k >= 1 , the class of k(+) -real face graphs is not included in any family of beyond-planar graphs with hereditary property.

Product structure theorems are a collection of recent results that have been used to resolve a number of longstanding open problems on planar graphs and related graph classes. One particularly useful version states that every planar graph $G$ is contained in the strong product of a $3$-tree $H$, a path $P$, and a $3$-cycle $K_3$; written as $G\subseteq H\boxtimes P\boxtimes K_3$. A number of researchers have asked if this theorem can be strengthened so that the maximum degree in $H$ can be bounded by a function of the maximum degree in $G$. We show that no such strengthening is possible. Specifically, we describe an infinite family $\mathcal{G}$ of planar graphs of maximum degree $5$ such that, if an $n$-vertex member $G$ of $\mathcal{G}$ is isomorphic to a subgraph of $H\boxtimes P\boxtimes K_c$ where $P$ is a path and $H$ is a graph of maximum degree $\Delta$ and treewidth $t$, then $t\Delta c \ge 2^{\Omega(\sqrt{\log\log n})}$.

LetTbe a tree ontvertices. We prove that for every positive integerkand every graphG, eitherGcontainskpairwise vertex-disjoint subgraphs each having aTminor, or there exists a setXof at mostt(k-1)vertices ofGsuch thatG-Xhas noTminor. The bound on the size ofXis best possible and improveson an earlierf(t)kbound proved by Fiorini, Joret, and Wood (2013) with some fast-growing functionf(t).Moreover, our proof is short and simple.

We prove that for every tree $T$ of radius $h$, there is an integer $c$ such that every $T$-minor-free graph is contained in $H\boxtimes K_c$ for some graph $H$ with pathwidth at most $2h-1$. This is a qualitative strengthening of the Excluded Tree Minor Theorem of Robertson and Seymour (GM I). We show that radius is the right parameter to consider in this setting, and $2h-1$ is the best possible bound.