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David R. Wood

Publications and source records attributed to David R. Wood.

At least 163 records · Page 9Linked to original sources
arXiv2012

Decompositions of Complete Multipartite Graphs into Complete Graphs

Let $k\geq\ell\geq1$ and $n\geq 1$ be integers. Let $G(k,n)$ be the complete $k$-partite graph with $n$ vertices in each colour class. An $\ell$-decomposition of $G(k,n)$ is a set $X$ of copies of $K_k$ in $G(k,n)$ such that each copy of $K_\ell$ in $G(k,n)$ is a subgraph of exactly one copy of $K_k$ in $X$. This paper asks: when does $G(k,n)$ have an $\ell$-decomposition? The answer is well known for the $\ell=2$ case. In particular, $G(k,n)$ has a 2-decomposition if and only if there exists $k-2$ mutually orthogonal Latin squares of order $n$. For general $\ell$, we prove that $G(k,n)$ has an $\ell$-decomposition if and only if there are $k-\ell$ Latin cubes of dimension $\ell$ and order $n$, with an additional property that we call mutually invertible. This property is stronger than being mutually orthogonal. An $\ell$-decomposition of $G(k,n)$ is then constructed whenever no prime less than $k$ divides $n$.

math.CO↗
arXiv2012

Nonrepetitive Colourings of Planar Graphs with $O(\log n)$ Colours

A vertex colouring of a graph is \emph{nonrepetitive} if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The \emph{nonrepetitive chromatic number} of a graph $G$ is the minimum integer $k$ such that $G$ has a nonrepetitive $k$-colouring. Whether planar graphs have bounded nonrepetitive chromatic number is one of the most important open problems in the field. Despite this, the best known upper bound is $O(\sqrt{n})$ for $n$-vertex planar graphs. We prove a $O(\log n)$ upper bound.

math.CO↗
arXiv2012

Cliques in Odd-Minor-Free Graphs

This paper is about: (1) bounds on the number of cliques in a graph in a particular class, and (2) algorithms for listing all cliques in a graph. We present a simple algorithm that lists all cliques in an $n$-vertex graph in O(n) time per clique. For O(1)-degenerate graphs, such as graphs excluding a fixed minor, we describe a O(n) time algorithm for listing all cliques. We prove that graphs excluding a fixed odd-minor have $O(n^2)$ cliques (which is tight), and conclude a $O(n^3)$ time algorithm for listing all cliques.

math.CO↗
arXiv2011

Disproof of the List Hadwiger Conjecture

The List Hadwiger Conjecture asserts that every $K_t$-minor-free graph is $t$-choosable. We disprove this conjecture by constructing a $K_{3t+2}$-minor-free graph that is not $4t$-choosable for every integer $t\geq 1$.

math.CO↗
arXiv2011

Clique Minors in Cartesian Products of Graphs

A "clique minor" in a graph G can be thought of as a set of connected subgraphs in G that are pairwise disjoint and pairwise adjacent. The "Hadwiger number" h(G) is the maximum cardinality of a clique minor in G. This paper studies clique minors in the Cartesian product G*H. Our main result is a rough structural characterisation theorem for Cartesian products with bounded Hadwiger number. It implies that if the product of two sufficiently large graphs has bounded Hadwiger number then it is one of the following graphs: - a planar grid with a vortex of bounded width in the outerface, - a cylindrical grid with a vortex of bounded width in each of the two `big' faces, or - a toroidal grid. Motivation for studying the Hadwiger number of a graph includes Hadwiger's Conjecture, which states that the chromatic number chi(G) <= h(G). It is open whether Hadwiger's Conjecture holds for every Cartesian product. We prove that if |V(H)|-1 >= chi(G) >= chi(H) then Hadwiger's Conjecture holds for G*H. On the other hand, we prove that Hadwiger's Conjecture holds for all Cartesian products if and only if it holds for all G * K_2. We then show that h(G * K_2) is tied to the treewidth of G. We also develop connections with pseudoachromatic colourings and connected dominating sets that imply near-tight bounds on the Hadwiger number of grid graphs (Cartesian products of paths) and Hamming graphs (Cartesian products of cliques).

math.CO↗
arXiv2011

Colouring the Triangles Determined by a Point Set

Let P be a set of n points in general position in the plane. We study the chromatic number of the intersection graph of the open triangles determined by P. It is known that this chromatic number is at least n^3/27+O(n^2), and if P is in convex position, the answer is n^3/24+O(n^2). We prove that for arbitrary P, the chromatic number is at most n^3/19.259+O(n^2).

math.CO↗
arXiv2011

Nordhaus-Gaddum for Treewidth

We prove that for every graph $G$ with $n$ vertices, the treewidth of $G$ plus the treewidth of the complement of $G$ is at least $n-2$. This bound is tight.

math.CO↗
arXiv2011

On the connectivity of visibility graphs

The visibility graph of a finite set of points in the plane has the points as vertices and an edge between two vertices if the line segment between them contains no other points. This paper establishes bounds on the edge- and vertex-connectivity of visibility graphs. Unless all its vertices are collinear, a visibility graph has diameter at most 2, and so it follows by a result of Plesník (1975) that its edge-connectivity equals its minimum degree. We strengthen the result of Plesník by showing that for any two vertices v and w in a graph of diameter 2, if deg(v) <= deg(w) then there exist deg(v) edge-disjoint vw-paths of length at most 4. Furthermore, we find that in visibility graphs every minimum edge cut is the set of edges incident to a vertex of minimum degree. For vertex-connectivity, we prove that every visibility graph with n vertices and at most l collinear vertices has connectivity at least (n-1)/(l-1), which is tight. We also prove the qualitatively stronger result that the vertex-connectivity is at least half the minimum degree. Finally, in the case that l=4 we improve this bound to two thirds of the minimum degree.

math.CO↗
arXiv2011

Treewidth of Cartesian Products of Highly Connected Graphs

The following theorem is proved: For all $k$-connected graphs $G$ and $H$ each with at least $n$ vertices, the treewidth of the cartesian product of $G$ and $H$ is at least $k(n -2k+2)-1$. For $n\gg k$ this lower bound is asymptotically tight for particular graphs $G$ and $H$. This theorem generalises a well known result about the treewidth of planar grid graphs.

math.CO↗
arXiv2011

The chromatic number of the convex segment disjointness graph

Let $P$ be a set of $n$ points in general and convex position in the plane. Let $D_n$ be the graph whose vertex set is the set of all line segments with endpoints in $P$, where disjoint segments are adjacent. The chromatic number of this graph was first studied by Araujo et al. [\emph{CGTA}, 2005]. The previous best bounds are $\frac{3n}{4}\leqχ(D_n) <n-\sqrt{\frac{n}{2}}$ (ignoring lower order terms). In this paper we improve the lower bound to $χ(D_n)\geq n-\sqrt{2n}$, to conclude a near-tight bound on $χ(D_n)$.

math.CO↗
arXiv2011

Polynomial treewidth forces a large grid-like-minor

Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an $\ell\times\ell$ grid minor is exponential in $\ell$. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A \emph{grid-like-minor of order} $\ell$ in a graph $G$ is a set of paths in $G$ whose intersection graph is bipartite and contains a $K_{\ell}$-minor. For example, the rows and columns of the $\ell\times\ell$ grid are a grid-like-minor of order $\ell+1$. We prove that polynomial treewidth forces a large grid-like-minor. In particular, every graph with treewidth at least $c\ell^4\sqrt{\log\ell}$ has a grid-like-minor of order $\ell$. As an application of this result, we prove that the cartesian product $G\square K_2$ contains a $K_{\ell}$-minor whenever $G$ has treewidth at least $c\ell^4\sqrt{\log\ell}$.

math.CO↗
arXiv2011

The maximum number of cliques in a graph embedded in a surface

This paper studies the following question: Given a surface $Σ$ and an integer $n$, what is the maximum number of cliques in an $n$-vertex graph embeddable in $Σ$? We characterise the extremal graphs for this question, and prove that the answer is between $8(n-ω)+2^ω$ and $8n+{3/2} 2^ω+o(2^ω)$, where $ω$ is the maximum integer such that the complete graph $K_ω$ embeds in $Σ$. For the surfaces $\mathbb{S}_0$, $\mathbb{S}_1$, $\mathbb{S}_2$, $\mathbb{N}_1$, $\mathbb{N}_2$, $\mathbb{N}_3$ and $\mathbb{N}_4$ we establish an exact answer.

math.CO↗
arXiv2011

Rooted $K_4$-Minors

Let $a,b,c,d$ be four vertices in a graph $G$. A \emph{$K_4$-minor rooted} at $a,b,c,d$ consists of four pairwise-disjoint pairwise-adjacent connected subgraphs of $G$, respectively containing $a,b,c,d$. We characterise precisely when $G$ contains a $K_4$-minor rooted at $a,b,c,d$ by describing six classes of obstructions, which are the edge-maximal graphs containing no $K_4$-minor rooted at $a,b,c,d$. The following two special cases illustrate the full characterisation: (1) A 4-connected non-planar graph contains a $K_4$-minor rooted at $a,b,c,d$ for every choice of $a,b,c,d$. (2) A 3-connected planar graph contains a $K_4$-minor rooted at $a,b,c,d$ if and only if $a,b,c,d$ are not on a single face.

math.CO↗
arXiv2010

Thomassen's Choosability Argument Revisited

Thomassen (1994) proved that every planar graph is 5-choosable. This result was generalised by {Š}krekovski (1998) and He et al. (2008), who proved that every $K_5$-minor-free graph is 5-choosable. Both proofs rely on the characterisation of $K_5$-minor-free graphs due to Wagner (1937). 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.

math.CO↗
arXiv2010

Partitions and Coverings of Trees by Bounded-Degree Subtrees

This paper addresses the following questions for a given tree $T$ and integer $d\geq2$: (1) What is the minimum number of degree-$d$ subtrees that partition $E(T)$? (2) What is the minimum number of degree-$d$ subtrees that cover $E(T)$? We answer the first question by providing an explicit formula for the minimum number of subtrees, and we describe a linear time algorithm that finds the corresponding partition. For the second question, we present a polynomial time algorithm that computes a minimum covering. We then establish a tight bound on the number of subtrees in coverings of trees with given maximum degree and pathwidth. Our results show that pathwidth is the right parameter to consider when studying coverings of trees by degree-3 subtrees. We briefly consider coverings of general graphs by connected subgraphs of bounded degree.

math.CO↗
arXiv2010

Proximity Drawings of High-Degree Trees

A drawing of a given (abstract) tree that is a minimum spanning tree of the vertex set is considered aesthetically pleasing. However, such a drawing can only exist if the tree has maximum degree at most 6. What can be said for trees of higher degree? We approach this question by supposing that a partition or covering of the tree by subtrees of bounded degree is given. Then we show that if the partition or covering satisfies some natural properties, then there is a drawing of the entire tree such that each of the given subtrees is drawn as a minimum spanning tree of its vertex set.

cs.CG↗
arXiv2010

The Big-Line-Big-Clique Conjecture is False for Infinite Point Sets

The big-line-big-clique conjecture states that for all $k,\ell\geq2$ there is an integer $n$ such that every finite set of at least $n$ points in the plane contains $\ell$ collinear points or $k$ pairwise visible points. We show that this conjecture is false for infinite point sets, by constructing a countably infinite point set with no 4 collinear points and no 3 pairwise visible points.

math.CO↗