arXiv subjects
Publications and source records attributed to Bruce Reed.
We show that Maker wins the Maker-Breaker perfect matching game in $\frac{n}{2}+o(n)$ turns when the bias is at least $\frac{n}{\log{n}}-\frac{f(n)n}{(\log{n})^{5/4}}$, for any $f$ going to infinity with $n$ and $n$ sufficiently large (in terms of $f$).
We show that there is a constant C such that for any $b<\frac{n}{\ln{n}}-\frac{Cn}{(\ln{n})^{3/2}}$, Maker wins the Maker-Breaker Hamilton cycle game in $n+\frac{Cn}{\sqrt{\ln{n}}}$ steps.
Tree-chromatic number is a chromatic version of treewidth, where the cost of a bag in a tree-decomposition is measured by its chromatic number rather than its size. Path-chromatic number is defined analogously. These parameters were introduced by Seymour (JCTB 2016). In this paper, we survey all the known results on tree- and path-chromatic number and then present some new results and conjectures. In particular, we propose a version of Hadwiger's Conjecture for tree-chromatic number. As evidence that our conjecture may be more tractable than Hadwiger's Conjecture, we give a short proof that every $K_5$-minor-free graph has tree-chromatic number at most $4$, which avoids the Four Colour Theorem. We also present some hardness results and conjectures for computing tree- and path-chromatic number.
The game of cops and robbers is a well-known game played on graphs. In this paper we consider the straight-ahead orientations of 4-regular quadrangulations of the torus and the Klein bottle and we prove that their cop number is bounded by a constant. We also show that the cop number of every k-regularly oriented toroidal grid is at most 13.
We show that for sufficiently large $d$ and for $t\geq d+1$, there is a graph $G$ with average degree $(1-\varepsilon)λt \sqrt{\ln d}$ such that almost every graph $H$ with $t$ vertices and average degree $d$ is not a minor of $G$, where $λ=0.63817\dots$ is an explicitly defined constant. This generalises analogous results for complete graphs by Thomason (2001) and for general dense graphs by Myers and Thomason (2005). It also shows that an upper bound for sparse graphs by Reed and Wood (2016) is best possible up to a constant factor.
A well-known conjecture of Erdős and Sós states that every graph with average degree exceeding $m-1$ contains every tree with $m$ edges as a subgraph. We propose a variant of this conjecture, which states that every graph of maximum degree exceeding $m$ and minimum degree at least $\lfloor \frac{2m}{3}\rfloor$ contains every tree with $m$ edges. As evidence for our conjecture we show (i) for every $m$ there is a $g(m)$ such that the weakening of the conjecture obtained by replacing $m$ by $g(m)$ holds, and (ii) there is a $γ>0$ such that the weakening of the conjecture obtained by replacing $\lfloor \frac{2m}{3}\rfloor$ by $(1-γ)m$ holds.
A vertex u of a graph t-dominates a vertex v if there are at most t vertices different from u,v that are adjacent to v and not to u; and a graph is t-dominating if for every pair of distinct vertices, one of them t-dominates the other. Our main result says that if a graph is t-dominating, then it is close (in an appropriate sense) to being 0-dominating. We also show that an analogous statement for digraphs is false; and discuss some connections with the Erdos-Hajnal conjecture.
Random walks on dynamic graphs have received increasingly more attention from different academic communities over the last decade. Despite the relatively large literature, little is known about random walks that construct the graph where they walk while moving around. In this paper we study one of the simplest conceivable discrete time models of this kind, which works as follows: before every walker step, with probability $p$ a new leaf is added to the vertex currently occupied by the walker. The model grows trees and we call it the Bernoulli Growth Random Walk (BGRW). We show that the BGRW walker is transient and has a well-defined linear speed $c(p)>0$ for any $0<p\leq 1$. We also show that the tree as seen by the walker converges (in a suitable sense) to a random tree that is one-ended. Some natural open problems about this tree and variants of our model are collected at the end of the paper.
We consider decomposing a 3-connected planar graph $G$ using laminar separators of size three. We show how to find a maximal set of laminar 3-separators in such a graph in linear time. We also discuss how to find maximal laminar set of 3-separators from special families. For example we discuss non-trivial cuts, ie. cuts which split $G$ into two components of size at least two. For any vertex $v$, we also show how to find a maximal set of 3-separators disjoint from $v$ which are laminar and satisfy: every vertex in a separator $X$ has two neighbours not in the unique component of $G-X$ containing $v$. In all cases, we show how to construct a corresponding tree decomposition of adhesion three. Our new algorithms form an important component of recent methods for finding disjoint paths in nonplanar graphs.
A {\em string graph} is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of {\em almost all} string graphs on $n$ vertices can be partitioned into {\em five} cliques such that some pair of them is not connected by any edge ($n\rightarrow\infty$). We also show that every graph with the above property is an intersection graph of plane convex sets. As a corollary, we obtain that {\em almost all} string graphs on $n$ vertices are intersection graphs of plane convex sets.
We study the height of a spanning tree $T$ of a graph $G$ obtained by starting with a single vertex of $G$ and repeatedly selecting, uniformly at random, an edge of $G$ with exactly one endpoint in $T$ and adding this edge to $T$.
In 1977, Wegner conjectured that the chromatic number of the square of every planar graph $G$ with maximum degree $Δ\ge8$ is at most $\bigl\lfloor\frac32Δ\bigr\rfloor+1$. We show that it is at most $\frac32 Δ(1+o(1))$ (where the $o(1)$ is as $Δ\to+\infty$), and indeed that this is true for the list chromatic number and for more general classes of graphs.
For a fixed degree sequence $\mathcal{D}=(d_1,...,d_n)$, let $G(\mathcal{D})$ be a uniformly chosen (simple) graph on $\{1,...,n\}$ where the vertex $i$ has degree $d_i$. In this paper we determine whether $G(\mathcal{D})$ has a giant component with high probability, essentially imposing no conditions on $\mathcal{D}$. We simply insist that the sum of the degrees in $\mathcal{D}$ which are not 2 is at least $λ(n)$ for some function $λ$ going to infinity with $n$. This is a relatively minor technical condition, and when $\mathcal{D}$ does not satisfy it, both the probability that $G(\mathcal{D})$ has a giant component and the probability that $G(\mathcal{D})$ has no giant component are bounded away from $1$.
We prove that every graph with circumference at most $k$ is $O(\log k)$-colourable such that every monochromatic component has size at most $O(k)$. The $O(\log k)$ bound on the number of colours is best possible, even in the setting of colourings with bounded monochromatic degree.
Let $\mathcal{F}$ be a family of fixed graphs and let $d$ be large enough. For every $d$-regular graph $G$, we study the existence of a spanning $\mathcal{F}$-free subgraph of $G$ with large minimum degree. This problem is well-understood if $\mathcal{F}$ does not contain bipartite graphs. Here we provide asymptotically tight results for many families of bipartite graphs such as cycles or complete bipartite graphs.
An edge colouring of a graph $G$ is called acyclic if it is proper and every cycle contains at least three colours. We show that for every $\varepsilon>0$, there exists a $g=g(\varepsilon)$ such that if $G$ has girth at least $g$ then $G$ admits an acyclic edge colouring with at most $(1+\varepsilon)Δ$ colours.
In this paper we show how to combine two algorithmic techniques to obtain linear time algorithms for various optimization problems on graphs, and present a subroutine which will be useful in doing so. The first technique is iterative shrinking. In the first phase of an iterative shrinking algorithm, we construct a sequence of graphs of decreasing size $G_1,\ldots,G_\ell$ where $G_1$ is the initial input, $G_\ell$ is a graph on which the problem is easy, and $G_i$ is obtained from $G_{i+1}$ via some shrinking algorithm. In the second phase we work through the sequence in reverse, repeatedly constructing a solution for a graph from the solution for its successor. In an iterative compaction algorithm, we insist that the graphs decrease by a constant fraction of the entire graph. Another approach to solving optimization problems is to exploit the structural properties implied by the connectivity of the input graph. This approach can be used on graphs which are not highly connected by decomposing an input graph into its highly connected pieces, solving subproblems on these specially structured pieces and then combining their solutions. We combine these two techniques by developing compaction algorithms which when applied to the highly connected pieces preserve their connectivity properties. The structural properties this connectivity implies can be helpful both in finding further compactions in later iterations and when we are manipulating solutions in the second phase of an iterative compaction algorithm. To illustrate how this compaction algorithm can be used as a subroutine, we present a linear time algorithm that given four vertices $\{s_1,s_2,t_1,t_2\}$ of a graph $G$, either finds a pair of disjoint paths $P_1$ and $P_2$ of $G$ such that $P_i$ has endpoints $s_i$ and $t_i$, or returns a planar embedding of an auxiliary graph which shows that no such pair exists.
This paper addresses the following question for a given graph $H$: what is the minimum number $f(H)$ such that every graph with average degree at least $f(H)$ contains $H$ as a minor? Due to connections with Hadwiger's Conjecture, this question has been studied in depth when $H$ is a complete graph. Kostochka and Thomason independently proved that $f(K_t)=ct\sqrt{\ln t}$. More generally, Myers and Thomason determined $f(H)$ when $H$ has a super-linear number of edges. We focus on the case when $H$ has a linear number of edges. Our main result, which complements the result of Myers and Thomason, states that if $H$ has $t$ vertices and average degree $d$ at least some absolute constant, then $f(H)\leq 3.895\sqrt{\ln d}\,t$. Furthermore, motivated by the case when $H$ has small average degree, we prove that if $H$ has $t$ vertices and $q$ edges, then $f(H) \leq t+6.291q$ (where the coefficient of 1 in the $t$ term is best possible).