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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 $|V(G)| \le (1+\delta)|V(T)|$.

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)., Comment: 15 pages, zero figures

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., Comment: 18 pages

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 $\Delta$.

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.

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., Comment: 11 pages

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 $f$)., Comment: 13 pages, 1 figure

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., Comment: v2: revised following referees' comments

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., Comment: 11 pages, 0 figures

We show that for sufficiently large $d$ and for $t\geq d+1$, there is a graph $G$ with average degree $(1-\varepsilon)\lambda t \sqrt{\ln d}$ such that almost every graph $H$ with $t$ vertices and average degree $d$ is not a minor of $G$, where $\lambda=0.63817\dots$ 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.

In this paper and a companion paper, we prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as a subgraph. Our result confirms, for large $m$, an important special case of a recent conjecture by Havet, Reed, Stein, and Wood. The present paper already contains an approximate version of the result., Comment: 59 pages

We prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as a subgraph. Our result confirms, for large $m$, an important special case of a conjecture by Havet, Reed, Stein, and Wood. The present paper builds on the results of a companion paper in which we proved the statement for all trees having a vertex that is adjacent to many leaves., Comment: 31 pages

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 the chromatic number of the $n$-dimensional associahedron grows at most logarithmically with $n$, improving a bound from and proving a conjecture of Fabila-Monroy et al. (2009)., Comment: 11 pages

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., Comment: 25 pages, 10 figures. To appear in the Proceedings of the 30th ACM-SIAM Symposium on Discrete Algorithms (SODA 2019)

Student homelessness is on the rise, with more than 1.3 million homeless students identified during the 2013-14 school year. This is a 7 percent increase from the previous year and more than double the number of homeless students in 2006-07. As high as these numbers seem, they are almost certainly undercounts. Despite increasing numbers, these students--as well as the school liaisons and state coordinators who support them--report that student homelessness remains an invisible and extremely disruptive problem. Students experiencing homelessness struggle to stay in school, to perform well, and to form meaningful connections with peers and adults. Ultimately, they are much more likely to fall off track and eventually drop out of school more often than their non-homeless peers. This study: (1) provides an overview of existing research on homeless students; (2) sheds light on the challenges homeless students face and the supports they say they need to succeed; (3) reports on the challenges adults--local liaisons and state coordinators--face in trying to help homeless students; and (4) recommends changes in policy and practice at the school, community, state and national level to help homeless students get on a path to adult success. Three appendices are included: (1) Paths Forward and ESSA Implementation Comparisons; (2) Major Wins for Homeless Students in the Every Student Succeeds Act (ESSA); and (3) Frequently Used Terms.

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., Comment: This is the full version of a paper appearing in the proceedings of SoCG 2018

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