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 ). Equivalently, every n-vertex planar graph G has a set X of ) vertices such that G — X has bandwidth ). This result holds in the more general setting of graphs contained in the strong product of a bounded treewidth graph and a path, which includes bounded genus graphs, graphs excluding a fixed apex graph as a minor, and k-planar graphs for fixed k. These results are obtained using two 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 )/D ) vertices whose removal results in a graph with local density at most D. The second is a generalization of a method of Feige and Rao, that relates bandwidth and local density using volume-preserving Euclidean embeddings.

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 ⊠ Kc. 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 given graph as a minor.

Abstract. Alon, Seymour, and Thomas [ J. Amer. Math. Soc., 3 (1990), pp. 801–808] proved that every [Formula: see text]-vertex graph excluding [Formula: see text] as a minor has treewidth less than [Formula: see text]. Illingworth, Scott, and Wood [ Product Structure of Graphs with an Excluded Minor, preprint, arXiv:2104.06627 , 2022] recently refined this result by showing that every such graph is a subgraph of some graph with treewidth [Formula: see text], where each vertex is blown up by a complete graph of order [Formula: see text]. Solving an open problem of Illingworth, Scott, and Wood [2022], we prove that the treewidth bound can be reduced to 4 while keeping blowups of order [Formula: see text]. As an extension of the Lipton–Tarjan theorem, in the case of planar graphs, we show that the treewidth can be further reduced to 2, which is best possible. We generalize this result for [Formula: see text]-minor-free graphs, with blowups of order [Formula: see text]. This setting includes graphs embeddable on any fixed surface.

Abstract. This is a corrigendum for the article “Orthogonal Tree-Decompositions of Graphs” [SIAM J. Discrete Math. 32(2):839–863, 2018].

Abstract. A circle graph is an intersection graph of a set of chords of a circle. We describe the unavoidable induced subgraphs of circle graphs with large treewidth. This includes examples that are far from the “usual suspects.” Our results imply that treewidth and Hadwiger number are linearly tied on the class of circle graphs and that the unavoidable induced subgraphs of a vertex-minor-closed class with large treewidth are the usual suspects if and only if the class has bounded rank-width. Using the same tools, we also study the treewidth of graphs [Formula: see text] that have a circular drawing whose crossing graph is well-behaved in some way. In this setting, we show that if the crossing graph is [Formula: see text]-minor-free, then [Formula: see text] has treewidth at most [Formula: see text] and has no [Formula: see text]-topological minor. On the other hand, we show that there are graphs with arbitrarily large Hadwiger number that have circular drawings whose crossing graphs are 2-degenerate.

Abstract. The planar graph product structure theorem of Dujmović et al. [ J. ACM, 67 (2020), 22] states that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. This result has been the key tool to resolve important open problems regarding queue layouts, nonrepetitive colorings, centered colorings, and adjacency labeling schemes. In this paper, we extend this line of research by utilizing shallow minors to prove analogous product structure theorems for several beyond-planar graph classes. The key observation that drives our work is that many beyond-planar graphs can be described as a shallow minor of the strong product of a planar graph with a small complete graph. In particular, we show that powers of bounded degree planar graphs, [Formula: see text]-planar, [Formula: see text]-cluster planar, fan-planar, and [Formula: see text]-fan-bundle planar graphs have such a shallow-minor structure. Using a combination of old and new results, we deduce that these classes have bounded queue-number, bounded nonrepetitive chromatic number, polynomial [Formula: see text]-centered chromatic numbers, linear strong coloring numbers, and cubic weak coloring numbers. In addition, we show that [Formula: see text]-gap planar graphs have at least exponential local treewidth and, as a consequence, cannot be described as a subgraph of the strong product of a graph with bounded treewidth and a path.

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 $t \geqslant s$, 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 den 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 $t \geqslant s \geqslant 3$, 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.

Abstract. Assume [Formula: see text] is a partition of [Formula: see text]. A [Formula: see text]-list assignment of [Formula: see text] is a [Formula: see text]-list assignment [Formula: see text] of [Formula: see text] such that the color set [Formula: see text] can be partitioned into [Formula: see text] sets [Formula: see text] such that for each [Formula: see text] and each vertex [Formula: see text] of [Formula: see text], [Formula: see text]. We say [Formula: see text] is [Formula: see text] -choosable if [Formula: see text] is [Formula: see text]-colorable for any [Formula: see text]-list assignment [Formula: see text] of [Formula: see text]. The concept of [Formula: see text]-choosability is a refinement of choosability that puts [Formula: see text]-choosability and [Formula: see text]-colorability in the same framework. If [Formula: see text] is close to [Formula: see text], then [Formula: see text]-choosability is close to [Formula: see text]-colorability; if [Formula: see text] is close to 1, then [Formula: see text]-choosability is close to [Formula: see text]-choosability. This paper studies Hadwiger’s conjecture in the context of [Formula: see text]-choosability. Hadwiger’s conjecture is equivalent to saying that every [Formula: see text]-minor-free graph is [Formula: see text]-choosable for any positive integer [Formula: see text], where [Formula: see text] is the multiset consisting of [Formula: see text] copies of 1. We prove that for [Formula: see text], for any partition [Formula: see text] of [Formula: see text] other than [Formula: see text], there is a [Formula: see text]-minor-free graph [Formula: see text] that is not [Formula: see text]-choosable. We then construct several types of [Formula: see text]-minor-free graphs that are not [Formula: see text]-choosable, where [Formula: see text] gets larger as [Formula: see text] gets larger. In particular, for any [Formula: see text] and any [Formula: see text], there exists [Formula: see text] such that for any [Formula: see text], for any partition [Formula: see text] of [Formula: see text] with [Formula: see text], there is a [Formula: see text]-minor-free graph that is not [Formula: see text]-choosable. The [Formula: see text] case of this result was recently proved by Steiner, and our proof uses a similar argument. We also generalise this result to [Formula: see text]-list coloring.

The Lovász Local Lemma is a powerful probabilistic technique for proving the existence of combinatorial objects. It is especially useful for coloring graphs and hypergraphs with bounded maximum degree. This paper presents a general theorem for coloring hypergraphs that in many instances matches or slightly improves upon the bounds obtained using the Lovász Local Lemma. Moreover, the theorem directly shows that there are exponentially many colorings. The elementary and self-contained proof is inspired by a recent result for nonrepetitive colorings by Rosenfeld [ Electron. J. Combin., 27 (2020), P3.43]. We apply our general theorem in the settings of proper hypergraph coloring, proper graph coloring, independent transversals, star coloring, nonrepetitive coloring, frugal coloring, Ramsey number lower bounds, and $k$-SAT.

A graph $G$ is Ramsey for a graph $H$ if every 2-coloring of the edges of $G$ contains a monochromatic copy of $H$. We consider the following question: if $H$ has bounded treewidth, is there a “sparse” graph $G$ that is Ramsey for $H$? Two notions of sparsity are considered. Firstly, we show that if the maximum degree and treewidth of $H$ are bounded, then there is a graph $G$ with $O(|V(H)|)$ edges that is Ramsey for $H$. This was previously only known for the smaller class of graphs $H$ with bounded bandwidth. On the other hand, we prove that in general the treewidth of a graph $G$ that is Ramsey for $H$ cannot be bounded in terms of the treewidth of $H$ alone. In fact, the latter statement is true even if the treewidth is replaced by the degeneracy and $H$ is a tree.

We show that planar graphs have bounded queue-number, thus proving a conjecture of Heath et al. [66] from 1992. The key to the proof is a new structural tool called layered partitions , and the result that every planar graph has a vertex-partition and a layering, such that each part has a bounded number of vertices in each layer, and the quotient graph has bounded treewidth. This result generalises for graphs of bounded Euler genus. Moreover, we prove that every graph in a minor-closed class has such a layered partition if and only if the class excludes some apex graph. Building on this work and using the graph minor structure theorem, we prove that every proper minor-closed class of graphs has bounded queue-number. Layered partitions have strong connections to other topics, including the following two examples. First, they can be interpreted in terms of strong products. We show that every planar graph is a subgraph of the strong product of a path with some graph of bounded treewidth. Similar statements hold for all proper minor-closed classes. Second, we give a simple proof of the result by DeVos et al. [31] that graphs in a proper minor-closed class have low treewidth colourings.

We prove that a minor-closed class of graphs has bounded layered pathwidth if and only if some apex-forest is not in the class. This generalizes a theorem of Robertson and Seymour, which says that a minor-closed class of graphs has bounded pathwidth if and only if some forest is not in the class.

We show that planar graphs have bounded queue-number, thus proving a conjecture of Heath, Leighton and Rosenberg from 1992. The key to the proof is a new structural tool called layered partitions, and the result that every planar graph has a vertex-partition and a layering, such that each part has a bounded number of vertices in each layer, and the quotient graph has bounded treewidth. This result generalises for graphs of bounded Euler genus. Moreover, we prove that every graph in a minor-closed class has such a layered partition if and only if the class excludes some apex graph. Building on this work and using the graph minor structure theorem, we prove that every proper minor-closed class of graphs has bounded queue-number. Layered partitions can be interpreted in terms of strong products. We show that every planar graph is a subgraph of the strong product of a path with some graph of bounded treewidth. Similar statements hold for all proper minor-closed classes.

This paper studies graphs that have two tree decompositions with the property that every bag from the first decomposition has a bounded-size intersection with every bag from the second decomposition. We show that every graph in each of the following classes has a tree decomposition and a linear-sized path decomposition with bounded intersections: (1) every proper minor-closed class, (2) string graphs with a linear number of crossings in a fixed surface, (3) graphs with linear crossing number in a fixed surface. Here “linear size” means that the total size of the bags in the path decomposition is $O(n)$ for $n$-vertex graphs. We then show that every $n$-vertex graph that has a tree decomposition and a linear-sized path decomposition with bounded intersections has $O(\sqrt{n})$ treewidth. As a corollary, we conclude a new lower bound on the crossing number of a graph in terms of its treewidth. Finally, we consider graph classes that have two path decompositions with bounded intersections. Trees and outerplanar graphs have this property. But for the next most simple class, series parallel graphs, we show that no such result holds.

The following is a corrigendum to [ Orthogonal tree decompositions of graphs, SIAM J. Discrete Math., 32 (2018), pp. 839--863].

We prove that every connected graph $G$ with $m$ edges contains a set $X$ of at most $\frac{3}{16}(m + 1)$ vertices such that $G-X$ has no $K_4$ minor, or, equivalently, has treewidth at most 2. This bound is best possible. Connectivity is essential: If $G$ is not connected, then only a bound of $\frac{1}{5}m$ can be guaranteed.

An anagram is a word of the form $WP$ where $W$ is a non-empty word and $P$ is a permutation of $W$. A vertex coloring of a graph is anagram-free if no subpath of the graph is an anagram. Anagram-free graph coloring was independently introduced by Kamčev, Łuczak, and Sudakov [ Combin. Probab. Comput., 27 (2018), pp. 623--642] and ourselves [ Electron. J. Combin., 25 (2018), pp. 2--20]. In this paper we introduce the study of anagram-free colorings of graph subdivisions. We show that every graph has an anagram-free 8-colorable subdivision. The number of division vertices per edge is exponential in the number of edges. For trees, we construct anagram-free $10$-colorable subdivisions with fewer division vertices per edge. Conversely, we prove lower bounds, in terms of division vertices per edge, on the anagram-free chromatic number for subdivisions of the complete graph and subdivisions of complete trees of bounded degree.

We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an $n$-vertex graph embedded on a surface of genus $g$ with at most $k$ crossings per edge has treewidth $O(\sqrt{(g+1)(k+1)n})$ and layered treewidth $O((g+1)k)$ and that these bounds are tight up to a constant factor. In the special case of $g=0$, so-called $k$-planar graphs, the treewidth bound is $O(\sqrt{(k+1)n})$, which is tight and improves upon a known $O((k+1)^{3/4}n^{1/2})$ bound. Analogous results are proved for map graphs defined with respect to any surface. Finally, we show that for $g

We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an n-vertex graph embedded on a surface of genus g with at most k crossings per edge has treewidth $$O(\sqrt{(g+1)(k+1)n})$$ and layered treewidth $$O((g+1)k)$$ , and that these bounds are tight up to a constant factor. As a special case, the k-planar graphs with n vertices have treewidth $$O(\sqrt{(k+1)n})$$ and layered treewidth $$O(k+1)$$ , which are tight bounds that improve a previously known $$O((k+1)^{3/4}n^{1/2})$$ treewidth bound. Additionally, we show that for $$g

An empty pentagon in a point set $P$ in the plane is a set of five points in $P$ in strictly convex position with no other point of $P$ in their convex hull. We prove that every finite set of at least $328\ell^2$ points in the plane contains an empty pentagon or $\ell$ collinear points. This is optimal up to a constant factor since the $(\ell -1)\times(\ell-1)$ square lattice contains no empty pentagon and no $\ell$ collinear points. The previous best known bound was doubly exponential.

We generalize a result of Corrádi and Hajnal and show that every graph with average degree at least $\frac{4}{3}kr$ contains $k$ vertex disjoint cycles, each of order at least $r$, as long as $k \geq 6$. This bound is sharp when r=3.

Graph separators are a ubiquitous tool in graph theory and computer science. However, in some applications, their usefulness is limited by the fact that the separator can be as large as Ω(√n) in graphs with n vertices. This is the case for planar graphs, and more generally, for proper minor-closed families. We study a special type of graph separator, called a layered separator, which may have linear size in n, but has bounded size with respect to a different measure, called the breadth. We prove that a wide class of graphs admit layered separators of bounded breadth, including graphs of bounded Euler genus. We use layered separators to prove Õ(log n) bounds for a number of problems where O(√n) was a long standing previous best bound. This includes the nonrepetitive chromatic number and queue-number of graphs with bounded Euler genus. We extend these results to all proper minor-closed families, with a O(log n) bound on the nonrepetitive chromatic number, and a logO(1)n bound on the queue-number. Only for planar graphs were logO(1)n bounds previously known. Our results imply that every graph from a proper minor-closed class has a 3-dimensional grid drawing with n logO(1)n volume, whereas the previous best bound was O(n3/2). Readers interested in the full details should consult arXiv:1302.0304 and arXiv:1306.1595, rather than the current extended abstract.

Let $g(t)$ be the minimum number such that every graph $G$ with average degree $d(G) \geq g(t)$ contains a $K_{t}$-minor. Such a function is known to exist, as originally shown by Mader. Kostochka and Thomason independently proved that $g(t) \in \Theta(t\sqrt{\log t})$. This paper shows that for all fixed $\epsilon > 0$ and fixed sufficiently large $t \geq t(\epsilon)$, if $d(G) \geq (2+\epsilon)g(t)$, then we can find this $K_{t}$-minor in linear time. This improves a previous result by Reed and Wood who gave a linear-time algorithm when $d(G) \geq 2^{t-2}$.

Let $f(n,\ell)$ be the maximum integer such that every set of $n$ points in the plane with at most $\ell$ collinear contains a subset of $f(n,\ell)$ points with no three collinear. First we prove that if $\ell\leqslant O(\sqrt{n})$, then $f(n,\ell)\geqslant\Omega(\sqrt{n/\ln\ell})$. Second we prove that if $\ell\leqslant O(n^{(1-\epsilon)/2})$, then $f(n,\ell)\geqslant\Omega(\sqrt{n\log_\ell n})$, which implies all previously known lower bounds on $f(n,\ell)$ and improves them when $\ell$ is not fixed. A more general problem is to consider subsets with at most $k$ collinear points in a point set with at most $\ell$ collinear. We also prove analogous results in this setting.

It is known that the First-Fit algorithm for partitioning a poset $P$ into chains uses relatively few chains when $P$ does not have two incomparable chains each of size $k$. In particular, if $P$ has width $w$, then Bosek, Krawczyk, and Szczypka [SIAM J. Discrete Math., 23 (2010), pp. 1992--1999], proved an upper bound of $ckw^{2}$ on the number of chains used by First-Fit for some constant $c$, while Joret and Milans [Order, 28 (2011), pp. 455--464] gave one of $ck^{2}w$. In this paper we prove an upper bound of the form $ckw$. This is most possible up to the value of $c$.

Thomassen (J. Combin. Theory Ser. B, 62 (1994), pp. 180–181) proved that every planar graph is 5-choosable. This result was generalized by Škrekovski (Discrete Math., 190 (1998), pp. 223–226) and He, Miao, and Shen (Discrete Math., 308 (2008), pp. 4024–4026), who proved that every $K_5$-minor-free graph is 5-choosable. Both proofs rely on the characterization of $K_5$-minor-free graphs due to Wagner (Math. Ann., 114 (1937), pp. 570–590). This paper proves the same result without using Wagner's structure theorem or even planar embeddings. Given that there is no structure theorem for graphs with no $K_6$-minor, we argue that this proof suggests a possible approach for attacking the Hadwiger Conjecture.

Let G be an n -vertex m -edge graph with weighted vertices. A pair of vertex sets A , B ⊆ V ( G ) is a 2/3 -separation of order | A ∩ B | if A ∪ B = V ( G ), there is no edge between A − B and B − A , and both A − B and B − A have weight at most 2/3 the total weight of G . Let ℓ ∈ Z + be fixed. Alon et al. [1990] presented an algorithm that in O ( n 1/2 m ) time, outputs either a K ℓ -minor of G , or a separation of G of order O ( n 1/2 ). Whether there is a O ( n + m )-time algorithm for this theorem was left as an open problem. In this article, we obtain a O ( n + m )-time algorithm at the expense of a O ( n 2/3 ) separator. Moreover, our algorithm exhibits a trade-off between time complexity and the order of the separator. In particular, for any given ϵ ∈ [0,1/2], our algorithm outputs either a K ℓ -minor of G , or a separation of G with order O ( n (2−ϵ)/3 in O ( n 1 + ϵ + m ) time. As an application we give a fast approximation algorithm for finding an independent set in a graph with no K ℓ-minor.

The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. Our main result is that every graph G that does not contain a fixed graph as a minor has crossing number O(Δn), where G has n vertices and maximum degree Δ. This dependence on n and Ø is best possible. This result answers an open question of Wood and Telle [New York J. Mathematics, 2007], who proved the best previous bound of O(Ø2n).

A set of vertices S resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. This paper studies the metric dimension of cartesian products $G\,\square\,H$. We prove that the metric dimension of $G\,\square\,G$ is tied in a strong sense to the minimum order of a so‐called doubly resolving set in G. Using bounds on the order of doubly resolving sets, we establish bounds on $G\,\square\,H$ for many examples of G and H. One of our main results is a family of graphs G with bounded metric dimension for which the metric dimension of $G\,\square\,G$ is unbounded.

Aqueue layout of a graph consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its queue-number. A three-dimensional (straight-line grid) drawing of a graph represents the vertices by points in $\mathbb{Z}^3$ and the edges by noncrossing line-segments. This paper contributes three main results: (1) It is proved that the minimum volume of a certain type of three-dimensional drawing of a graph G is closely related to the queue-number of G. In particular, if G is an n-vertex member of a proper minor-closed family of graphs (such as a planar graph), then G has a $\mathcal{O}(1) \times \mathcal{O}(1) \times \mathcal{O}(n)$ drawing if and only if G has a $\mathcal{O}(1)$ queue-number. (2) It is proved that the queue-number is bounded by the tree-width, thus resolving an open problem due to Ganley and Heath [Discrete Appl. Math., 109 (2001), pp. 215--221] and disproving a conjecture of Pemmaraju [Exploring the Powers of Stacks and Queues via Graph Layouts, Ph. D. thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA, 1992]. This result provides renewed hope for the positive resolution of a number of open problems in the theory of queue layouts. (3) It is proved that graphs of bounded tree-width have three-dimensional drawings with $\mathcal{O}(n)$ volume. This is the most general family of graphs known to admit three-dimensional drawings with $\mathcal{O}(n)$ volume. The proofs depend upon our results regarding track layouts and tree-partitions of graphs, which may be of independent interest.

In this paper, we consider hot-potato packet routing of one-to-one routing patterns on $n$-node trees. By applying a ‘charging argument’, we show that any greedy hot-potato algorithm routes a one-to-one routing pattern within $2(n-1)$ steps. On the other hand, a trivial lower bound suggests that at least $3n/2$ steps are required by any oblivious greedy algorithm. As the main contribution of the paper, we tighten the $2(n-1)$ upper bound by constructing (for all sufficiently large $n$) an elaborate one-to-one packet routing problem on an $n$-node tree for which an oblivious greedy hot-potato algorithm requires at least $2n-o(n)$ steps. This improved lower bound is also shown to be valid for the minimum-distance heuristic. For trees of maximum degree $d$, we establish a lower bound of $2((d-3)/(d-2))n-o(n)$ routing steps.