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Bruce Reed

Publications and source records attributed to Bruce Reed.

At least 19 recordsLinked to original sources
arXiv2026

The extremal cases of the Erd\H os--S\'os conjecture

The Erd\H os--S\'os conjecture states that every $n$-vertex graph $G$ with more than $(k-2)n/2$ edges contains every $k$-vertex tree. We solve the extremal cases of this conjecture, showing that for some fixed $\mu>0$, the conjecture holds for each $G$ that minimally satisfies the assumptions of the conjecture and has a subgraph~$H$ of minimum degree $\delta(H)\ge (1-\mu)k$. In our proof, we mainly have to deal with $H$ taking two different shapes: either $H$ is close to the complete graph $K_k$ or $H$ is close to the complete bipartite graph $K_{k,k}$.

math.CO↗
arXiv2026

The Erd\H os-S\'os conjecture in dense graphs

The Erd\H os--S\'os conjecture states that every $n$-vertex graph with more than $(k-2)n/2$ edges contains every $k$-vertex tree. We prove that for every $\gamma$ there is an $n_0$ such that for all $n\ge n_0$ and $k \ge \gamma n$ the conjecture holds. As a corollary of our result, we obtain a solution of a 51-year-old problem of Erd\H os and Graham on the multicolor Ramsey numbers of trees.

math.CO↗
arXiv2026

On Balance, To What Degree is Burr's Conjecture True?

For many trees $T$, the Ramsey number of $T$, denoted by ${\mathcal R}(T)$, is determined by the sizes of the partition classes in its unique bipartition. In 1976, Burr proved that when $T$ has partition classes of size $t_1$ and $t_2$ with $t_1 \le t_2$, the Ramsey number is at least $\max(2t_2-1,2t_1+t_2-1)$, and conjectured that this is tight. While counterexamples have been found for some pairs $(t_1, t_2)$, a main focus of research on this problem has been determining ratios $t_2/t_1$ or bounds on the maximum degree of $T$ for which Burr's bound is either exactly or asymptotically tight. We essentially resolve these questions for lopsided trees. Specifically, we show that (a) there are counterexamples whenever $t_2 \ge 2t_1$, with the order of magnitude of the difference between the largest Ramsey numbers and Burr's bound being $\max \left( t_1^2/t_2, \sqrt{t_1} \right)$, and (b) for $t_2 \ge 500 t_1$, Burr's bound is tight when $\Delta(T) \le t_2 - t_1$, but is off by at least $C \log t_2$ (even when $t_2 \ge 2 t_1$) when $\Delta(T) \gtrsim t_2 - t_1$. In particular, this shows that Burr's bound need not hold for $t$-vertex trees $T$ with $\Delta(T) \approx t/3$.

math.CO↗
arXiv2026

Diameters and mixing times for giant components of random graphs with given degrees

A sequence $D = \{d_1,...d_n\}$ is a feasible degree sequence if there is a graph on $\{1,...,n\}$ such that $i$ has degree $d_i$. For such a sequence, $G(D)$ is a graph chosen uniformly at random from those with the given degree sequence. We consider sequences $\{D_\ell\}_{\ell \geq 1}$ of feasible degree sequences which have a giant component. We show that with high probability this giant component is unique, and bound its diameter and the mixing time of the random walk on it. We also bound the size and diameter of the other components and show that many of these bounds are tight.

math.CO↗
arXiv2026

What is The Probability That A Random Graph With A Given Degree Sequence is Connected?

An $n$-tuple $D=(d(1),\dots,d(n))$ is a \emph{feasible degree sequence} if there is a graph on $\{1,\dots,n\}$ such that $i$ has degree $d(i)$. Any such graph will have $m=\sum_{i=1}^n d(i)/2$ edges. Letting $G(D)$ be a graph chosen uniformly from those with the given degree sequence, we upper-bound the probability that $G(D)$ is disconnected based on the number of vertices of degree $d$ for small $d$, and develop a powerful tool for proving such bounds. If there are any vertices of degree zero the probability $G$ is disconnected is $1$, so we assume there are no such vertices. Our results then imply that if there are $o(\sqrt{m})$ vertices of degree $1$ and $o(m)$ vertices of degree 2 then with high probability $G$ is connected, while if there are no vertices of degree 1 or 2 then the probability $G$ is disconnected is $O(\frac{n^4}{m^6})$.

math.PR↗
arXiv2025

The Global Structure of a Typical Graph Without $H$ as an Induced Subgraph when $H$ is a Cycle

One way to certify that a graph does not contain an induced cycle of length six is to provide a partition of its vertex set into (i) a stable set, and (ii) a graph containing no stable set of size three and no induced matching of size two. We show that almost every graph which does not contain a cycle of length six as an induced subgraph has such a certificate. We obtain similar characterizations of the structure of almost all graphs which contain no induced cycle of length $k$ for all even $k$ exceeding six. (Similar results were obtained for $k=3$ by Erdos, Kleitman, and Rothschild in 1976, for $k =4,5$ by Promel and Steger in 1991 and for odd $k$ exceeding 5 by Balogh and Butterfield in 2009.) We prove that a simiiar theorem for all $H$ holds up to the deletion of a set of $o(|V(G)|)$ vertices and ask for which $H$ the characterization holds fully.

math.CO↗
arXiv2025

Typical $T$-free graphs

We prove that for every tree $T$ which is not an edge, for almost every graph $G$ which does not contain $T$ as an induced subgraph, $V(G)$ has a partition into $\alpha(T)-1$ parts certifying this fact. Each part induces a graph which is $P_4$-free and has further properties which depend on $T$. As a consequence we obtain good bounds (often tight up to a constant factor) on the number of $T$-free graphs and show in a follow-up paper~\cite{RY} that almost every $T$-free graph $G$ has chromatic number equal to the size of its largest clique.

math.CO↗
arXiv2025

The asymptotic $\chi$-boundedness of hereditary families

A family ${\cal F}$ of graphs is asymptotically $\chi$-bounded with bounding function $f$ if almost every graph $G$ in the family satisfies $\chi(G) \le f(\omega(G))$. A graph is $H$-free if it does not contain $H$ as an induced subgraph. We ask which hereditary families are asymptotically $\chi$-bounded, and discuss some related questions. We show that for every tree $T$, almost all $T$-free graphs $G$ satisfy $\chi(G)=\omega(G)$. We show that for every cycle $C_k$ except $C_6$, almost every $C_k$-free graph $G$ satisfies $\chi(G) = \omega(G)$. We show that the $C_6$-free graphs are asymptotically $\chi$-bounded with bounding function $f(w)=(1+o(1))\frac{w^2}{\log w}$.

math.CO↗
arXiv2025

Bounds on treewidth via excluding disjoint unions of cycles

One of the fundamental results in graph minor theory is that for every planar graph~$H$, there is a minimum integer~$f(H)$ such that graphs with no minor isomorphic to~$H$ have treewidth at most~$f(H)$. The best known bound for an arbitrary planar $H$ is ${O(|V(H)|^9\operatorname{poly~log} |V(H)|)}$. We show that if $H$ is the disjoint union of cycles, then $f(H)$ is $O(|V(H)|\log^2 |V(H)|)$, which is a $\log|V(H)|$ factor away being optimal.

math.CO↗
arXiv2024

Embedding Nearly Spanning Trees

The Erd\H{o}s-S\'os Conjecture states that every graph with average degree exceeding $k-1$ contains every tree with $k$ edges as a subgraph. We prove that there are $\delta>0$ and $k_0\in\mathbb N$ such that the conjecture holds for every tree $T$ with $k \ge k_0$ edges and every graph $G$ with $|V(G)| \le (1+\delta)|V(T)|$.

math.CO↗
arXiv2024

Vertex Ranking of Degenerate Graphs

An $\ell$-vertex-ranking of a graph $G$ is a colouring of the vertices of $G$ with integer colours so that in any connected subgraph $H$ of $G$ with diameter at most $\ell$, there is a vertex in $H$ whose colour is larger than that of every other vertex in $H$. The $\ell$-vertex-ranking number, $\chi_{\ell-\mathrm{vr}}(G)$, of $G$ is the minimum integer $k$ such that $G$ has an $\ell$-vertex-ranking using $k$ colours. We prove that, for any fixed $d$ and $\ell$, every $d$-degenerate $n$-vertex graph $G$ satisfies $\chi_{\ell-\mathrm{vr}}(G)= O(n^{1-2/(\ell+1)}\log n)$ if $\ell$ is even and $\chi_{\ell-\mathrm{vr}}(G)= O(n^{1-2/\ell}\log n)$ if $\ell$ is odd. The case $\ell=2$ resolves (up to the $\log n$ factor) an open problem posed by \citet{karpas.neiman.ea:on} and the cases $\ell\in\{2,3\}$ are asymptotically optimal (up to the $\log n$ factor).

math.CO↗
arXiv2024

Linear bounds on treewidth in terms of excluded planar minors

One of the fundamental results in graph minor theory is that for every planar graph $H$, there is a minimum integer $f(H)$ such that graphs with no minor isomorphic to $H$ have treewidth at most $f(H)$. A lower bound for ${f(H)}$ can be obtained by considering the maximum integer $k$ such that $H$ contains $k$ vertex-disjoint cycles. There exists a graph of treewidth ${\Omega(k\log k)}$ which does not contain $k$ vertex-disjoint cycles, from which it follows that ${f(H) = \Omega(k\log k)}$. In particular, if ${f(H)}$ is linear in ${\lvert{V(H)}\rvert}$ for graphs $H$ from a subclass of planar graphs, it is necessary that $n$-vertex graphs from the class contain at most ${O(n/\log(n))}$ vertex-disjoint cycles. We ask whether this is also a sufficient condition, and demonstrate that this is true for classes of planar graphs with bounded component size. For an $n$-vertex graph $H$ which is a disjoint union of $r$ cycles, we show that ${f(H) \leq 3n/2 + O(r^2 \log r)}$, and improve this to ${f(H) \leq n + O(\sqrt{n})}$ when ${r = 2}$. In particular this bound is linear when ${r=O(\sqrt{n}/\log(n))}$. We present a linear bound for ${f(H)}$ when $H$ is a subdivision of an $r$-edge planar graph for any constant $r$. We also improve the best known bounds for ${f(H)}$ when $H$ is the wheel graph or the ${4 \times 4}$ grid, obtaining a bound of $160$ for the latter.

math.CO↗
arXiv2024

Peaceful Colourings

We introduce peaceful colourings, a variant of $h$-conflict free colourings. We call a colouring with no monochromatic edges $p$-peaceful if for each vertex $v$, there are at most $p$ neighbours of $v$ coloured with a colour appearing on another neighbour of $v$. An $h$-conflict-free colouring of a graph is a (vertex)-colouring with no monochromatic edges so that for every vertex $v$, the number of neighbours of $v$ which are coloured with a colour appearing on no other neighbour of $v$ is at least the minimum of $h$ and the degree of $v$. If $G$ is $\Delta$-regular then it has an $h$-conflict free colouring precisely if it has a $(\Delta-h)$-peaceful colouring. We focus on the minimum $p_\Delta$ of those $p$ for which every graph of maximum degree $\Delta$ has a $p$-peaceful colouring with $\Delta+1$ colours. We show that $p_\Delta > (1-\frac{1}{e}-o(1))\Delta$ and that for graphs of bounded codegree, $p_\Delta \leq (1-\frac{1}{e}+o(1))\Delta$. We ask if the latter result can be improved by dropping the bound on the codegree. As a partial result, we show that $p_\Delta \leq \frac{8000}{8001}\Delta$ for sufficiently large $\Delta$.

math.CO↗
arXiv2024

Asymptotically Optimal Proper Conflict-Free Colouring

A proper conflict-free colouring of a graph is a colouring of the vertices such that any two adjacent vertices receive different colours, and for every non-isolated vertex $v$, some colour appears exactly once on the neighbourhood of $v$. Caro, Petru\v{s}evski and \v{S}krekovski conjectured that every connected graph with maximum degree $\Delta \geq 3$ has a proper conflict-free colouring with at most $\Delta+1$ colours. This conjecture holds for $\Delta=3$ and remains open for $\Delta \geq 4$. In this paper we prove that this conjecture holds asymptotically; namely, every graph with maximum degree $\Delta$ has a proper conflict-free colouring with $(1+o(1))\Delta$ colours.

math.CO↗
arXiv2020

The Speed and Threshold of the Biased Perfect Matching Game

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 $f$).

math.CO↗
arXiv2020

Tight Bounds on the Clique Chromatic Number

The clique chromatic number of a graph is the minimum number of colours needed to colour its vertices so that no inclusion-wise maximal clique which is not an isolated vertex is monochromatic. We show that every graph of maximum degree $\Delta$ has clique chromatic number $O\left(\frac{\Delta}{\log~\Delta}\right)$. We obtain as a corollary that every $n$-vertex graph has clique chromatic number $O\left(\sqrt{\frac{n}{\log ~n}}\right)$. Both these results are tight.

math.CO↗
arXiv2020

Notes on Tree- and Path-chromatic Number

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.

math.CO↗