Colouring the Square of the Cartesian Product of Trees
We prove upper and lower bounds on the chromatic number of the square of the cartesian product of trees. The bounds are equal if each tree has even maximum degree.
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arXiv subjects
David R. Wood
Publications and source records attributed to David R. Wood.
Irreducible Triangulations are Small
A triangulation of a surface is \emph{irreducible} if there is no edge whose contraction produces another triangulation of the surface. We prove that every irreducible triangulation of a surface with Euler genus $g\geq1$ has at most $13g-4$ vertices. The best previous bound was $171g-72$.
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Blocking Coloured Point Sets
This paper studies problems related to visibility among points in the plane. A point $x$ \emph{blocks} two points $v$ and $w$ if $x$ is in the interior of the line segment $\bar{vw}$. A set of points $P$ is \emph{$k$-blocked} if each point in $P$ is assigned one of $k$ colours, such that distinct points $v,w\in P$ are assigned the same colour if and only if some other point in $P$ blocks $v$ and $w$. The focus of this paper is the conjecture that each $k$-blocked set has bounded size (as a function of $k$). Results in the literature imply that every 2-blocked set has at most 3 points, and every 3-blocked set has at most 6 points. We prove that every 4-blocked set has at most 12 points, and that this bound is tight. In fact, we characterise all sets $\{n_1,n_2,n_3,n_4\}$ such that some 4-blocked set has exactly $n_i$ points in the $i$-th colour class. Amongst other results, for infinitely many values of $k$, we construct $k$-blocked sets with $k^{1.79...}$ points.
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On Visibility and Blockers
This expository paper discusses some conjectures related to visibility and blockers for sets of points in the plane.
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On the book thickness of $k$-trees
Every $k$-tree has book thickness at most $k+1$, and this bound is best possible for all $k\geq3$. Vandenbussche et al. (2009) proved that every $k$-tree that has a smooth degree-3 tree decomposition with width $k$ has book thickness at most $k$. We prove this result is best possible for $k\geq 4$, by constructing a $k$-tree with book thickness $k+1$ that has a smooth degree-4 tree decomposition with width $k$. This solves an open problem of Vandenbussche et al. (2009)
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Token Graphs
For a graph $G$ and integer $k\geq1$, we define the token graph $F_k(G)$ to be the graph with vertex set all $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is a pair of adjacent vertices in $G$. Thus vertices of $F_k(G)$ correspond to configurations of $k$ indistinguishable tokens placed at distinct vertices of $G$, where two configurations are adjacent whenever one configuration can be reached from the other by moving one token along an edge from its current position to an unoccupied vertex. This paper introduces token graphs and studies some of their properties including: connectivity, diameter, cliques, chromatic number, Hamiltonian paths, and Cartesian products of token graphs.
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Every Large Point Set contains Many Collinear Points or an Empty Pentagon
We prove the following generalised empty pentagon theorem: for every integer $\ell \geq 2$, every sufficiently large set of points in the plane contains $\ell$ collinear points or an empty pentagon. As an application, we settle the next open case of the "big line or big clique" conjecture of Kára, Pór, and Wood [\emph{Discrete Comput. Geom.} 34(3):497--506, 2005].
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Characterisations and Examples of Graph Classes with Bounded Expansion
Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Nešetřil and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor. Several linear-time algorithms are known for bounded expansion classes (such as subgraph isomorphism testing), and they allow restricted homomorphism dualities, amongst other desirable properties. In this paper we establish two new characterisations of bounded expansion classes, one in terms of so-called topological parameters, the other in terms of controlling dense parts. The latter characterisation is then used to show that the notion of bounded expansion is compatible with Erdös-Rényi model of random graphs with constant average degree. In particular, we prove that for every fixed $d>0$, there exists a class with bounded expansion, such that a random graph of order $n$ and edge probability $d/n$ asymptotically almost surely belongs to the class. We then present several new examples of classes with bounded expansion that do not exclude some topological minor, and appear naturally in the context of graph drawing or graph colouring. In particular, we prove that the following classes have bounded expansion: graphs that can be drawn in the plane with a bounded number of crossings per edge, graphs with bounded stack number, graphs with bounded queue number, and graphs with bounded non-repetitive chromatic number. We also prove that graphs with `linear' crossing number are contained in a topologically-closed class, while graphs with bounded crossing number are contained in a minor-closed class.
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Graph Minors and Minimum Degree
Let $\mathcal{D}_k$ be the class of graphs for which every minor has minimum degree at most $k$. Then $\mathcal{D}_k$ is closed under taking minors. By the Robertson-Seymour graph minor theorem, $\mathcal{D}_k$ is characterised by a finite family of minor-minimal forbidden graphs, which we denote by $\widehat{\mathcal{D}}_k$. This paper discusses $\widehat{\mathcal{D}}_k$ and related topics. We obtain four main results: We prove that every $(k+1)$-regular graph with less than ${4/3}(k+2)$ vertices is in $\widehat{\mathcal{D}}_k$, and this bound is best possible. We characterise the graphs in $\widehat{\mathcal{D}}_{k+1}$ that can be obtained from a graph in $\widehat{\mathcal{D}}_k$ by adding one new vertex. For $k\leq 3$ every graph in $\widehat{\mathcal{D}}_k$ is $(k+1)$-connected, but for large $k$, we exhibit graphs in $\widehat{\mathcal{D}}_k$ with connectivity 1. In fact, we construct graphs in $\mathcal{D}_k$ with arbitrary block structure. We characterise the complete multipartite graphs in $\widehat{\mathcal{D}}_k$, and prove analogous characterisations with minimum degree replaced by connectivity, treewidth, or pathwidth.
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Contractibility and the Hadwiger Conjecture
Consider the following relaxation of the Hadwiger Conjecture: For each $t$ there exists $N_t$ such that every graph with no $K_t$-minor admits a vertex partition into $\ceil{αt+β}$ parts, such that each component of the subgraph induced by each part has at most $N_t$ vertices. The Hadwiger Conjecture corresponds to the case $α=1$, $β=-1$ and $N_t=1$. Kawarabayashi and Mohar [\emph{J. Combin. Theory Ser. B}, 2007] proved this relaxation with $α={31/2}$ and $β=0$ (and $N_t$ a huge function of $t$). This paper proves this relaxation with $α={7/2}$ and $β=-{3/2}$. The main ingredients in the proof are: (1) a list colouring argument due to Kawarabayashi and Mohar, (2) a recent result of Norine and Thomas that says that every sufficiently large $(t+1)$-connected graph contains a $K_t$-minor, and (3) a new sufficient condition for a graph to have a set of edges whose contraction increases the connectivity.
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Colourings of the Cartesian Product of Graphs and Multiplicative Sidon Sets
Let $F$ be a family of connected bipartite graphs, each with at least three vertices. A proper vertex colouring of a graph $G$ with no bichromatic subgraph in $F$ is $\F$-free. The $F$-free chromatic number $χ(G,F)$ of a graph $G$ is the minimum number of colours in an $F$-free colouring of $G$. For appropriate choices of $F$, several well-known types of colourings fit into this framework, including acyclic colourings, star colourings, and distance-2 colourings. This paper studies $F$-free colourings of the cartesian product of graphs. Let $H$ be the cartesian product of the graphs $G_1,G_2,...,G_d$. Our main result establishes an upper bound on the $F$-free chromatic number of $H$ in terms of the maximum $F$-free chromatic number of the $G_i$ and the following number-theoretic concept. A set $S$ of natural numbers is $k$-multiplicative Sidon if $ax=by$ implies $a=b$ and $x=y$ whenever $x,y\in S$ and $1\leq a,b\leq k$. Suppose that $χ(G_i,F)\leq k$ and $S$ is a $k$-multiplicative Sidon set of cardinality $d$. We prove that $χ(H,F) \leq 1+2k\cdot\max S$. We then prove that the maximum density of a $k$-multiplicative Sidon set is $Θ(1/\log k)$. It follows that $χ(H,F) \leq O(dk\log k)$. We illustrate the method with numerous examples, some of which generalise or improve upon existing results in the literature.
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Distinct Distances in Graph Drawings
The \emph{distance-number} of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the distance-number of trees, graphs with no $K^-_4$-minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distance-number of graphs with bounded degree. We prove that $n$-vertex graphs with bounded maximum degree and bounded treewidth have distance-number in $\mathcal{O}(\log n)$. To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth 2 and polynomial distance-number. Similarly, we prove that there exist graphs with maximum degree 5 and arbitrarily large distance-number. Moreover, as $Δ$ increases the existential lower bound on the distance-number of $Δ$-regular graphs tends to $Ω(n^{0.864138})$.
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Folding = Colouring
The foldings of a connected graph $G$ are defined as follows. First, $G$ is a folding of itself. Let $G'$ be a graph obtained from $G$ by identifying two vertices at distance 2 in $G$. Then every folding of $G'$ is a folding of $G$. The folding number of $G$ is the minimum order of a complete folding of $G$. Theorem: The folding number of every graph equals its chromatic number.
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Compatible Geometric Matchings
This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are \emph{compatible} if they have the same vertex set and their union is also non-crossing. Our first result states that for any two perfect matchings $M$ and $M'$ of the same set of $n$ points, for some $k\in\Oh{\log n}$, there is a sequence of perfect matchings $M=M_0,M_1,...,M_k=M'$, such that each $M_i$ is compatible with $M_{i+1}$. This improves the previous best bound of $k\leq n-2$. We then study the conjecture: \emph{every perfect matching with an even number of edges has an edge-disjoint compatible perfect matching}. We introduce a sequence of stronger conjectures that imply this conjecture, and prove the strongest of these conjectures in the case of perfect matchings that consist of vertical and horizontal segments. Finally, we prove that every perfect matching with $n$ edges has an edge-disjoint compatible matching with approximately $4n/5$ edges.
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On Tree-Partition-Width
A \emph{tree-partition} of a graph $G$ is a proper partition of its vertex set into `bags', such that identifying the vertices in each bag produces a forest. The \emph{tree-partition-width} of $G$ is the minimum number of vertices in a bag in a tree-partition of $G$. An anonymous referee of the paper by Ding and Oporowski [\emph{J. Graph Theory}, 1995] proved that every graph with tree-width $k\geq3$ and maximum degree $Δ\geq1$ has tree-partition-width at most $24kΔ$. We prove that this bound is within a constant factor of optimal. In particular, for all $k\geq3$ and for all sufficiently large $Δ$, we construct a graph with tree-width $k$, maximum degree $Δ$, and tree-partition-width at least $(\eighth-ε)kΔ$. Moreover, we slightly improve the upper bound to ${5/2}(k+1)({7/2}Δ-1)$ without the restriction that $k\geq3$.
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A polynomial bound for untangling geometric planar graphs
To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos [Discrete Comput. Geom., 2002] asked if every n-vertex geometric planar graph can be untangled while keeping at least n^εvertices fixed. We answer this question in the affirmative with ε=1/4. The previous best known bound was Ω((\log n / \log\log n)^{1/2}). We also consider untangling geometric trees. It is known that every n-vertex geometric tree can be untangled while keeping at least (n/3)^{1/2} vertices fixed, while the best upper bound was O(n\log n)^{2/3}. We answer a question of Spillner and Wolff [arXiv:0709.0170 2007] by closing this gap for untangling trees. In particular, we show that for infinitely many values of n, there is an n-vertex geometric tree that cannot be untangled while keeping more than 3(n^{1/2}-1) vertices fixed. Moreover, we improve the lower bound to (n/2)^{1/2}.
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Notes on Nonrepetitive Graph Colouring
A vertex colouring of a graph is \emph{nonrepetitive on paths} if there is no path $v_1,v_2,...,v_{2t}$ such that v_i and v_{t+i} receive the same colour for all i=1,2,...,t. We determine the maximum density of a graph that admits a k-colouring that is nonrepetitive on paths. We prove that every graph has a subdivision that admits a 4-colouring that is nonrepetitive on paths. The best previous bound was 5. We also study colourings that are nonrepetitive on walks, and provide a conjecture that would imply that every graph with maximum degree $Δ$ has a $f(Δ)$-colouring that is nonrepetitive on walks. We prove that every graph with treewidth k and maximum degree $Δ$ has a $O(kΔ)$-colouring that is nonrepetitive on paths, and a $O(kΔ^3)$-colouring that is nonrepetitive on walks.
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Planar Decompositions and the Crossing Number of Graphs with an Excluded Minor
Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar decomposition with bounded width and a linear number of bags. The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. We prove that planar decompositions are intimately related to the crossing number. In particular, a graph with bounded degree has linear crossing number if and only if it has a planar decomposition with bounded width and a linear number of bags. It follows from the above result about planar decompositions that every graph with bounded degree and an excluded minor has linear crossing number. Analogous results are proved for the convex and rectilinear crossing numbers. In particular, every graph with bounded degree and bounded tree-width has linear convex crossing number, and every $K_{3,3}$-minor-free graph with bounded degree has linear rectilinear crossing number.
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