We study the Localization game on locally finite graphs trees, where each of the countably many vertices have finite degree. In contrast to the finite case, we construct a locally finite tree with localization number n for any choice of positive integer n. Our examples have uncountably many ends, and we show that this is necessary by proving that locally finite trees with finitely or countably many ends have localization number at most 2. Finally, as is the case for finite graphs, we prove that any locally finite graph contains a subdivision where one cop can capture the robber.

Graph burning is a discrete-time process that models the spread of influence in a network. Vertices are either burning or unburned, and in each round, a burning vertex causes all of its neighbours to become burning before a new fire source is chosen to become burning. We introduce a variation of this process that incorporates an adversarial game played on a nested, growing sequence of graphs. Two players, Arsonist and Builder, play in turns: Builder adds a certain number of new unburned vertices and edges incident to these to create a larger graph, then every vertex neighbouring a burning vertex becomes burning, and finally Arsonist `burns' a new fire source. This process repeats forever. Arsonist is said to win if the limiting fraction of burning vertices tends to 1, while Builder is said to win if this fraction is bounded away from 1. The central question of this paper is determining if, given that Builder adds $f(n)$ vertices at turn $n$, either Arsonist or Builder has a winning strategy. In the case that $f(n)$ is asymptotically polynomial, we give threshold results for which player has a winning strategy.

Generalized Tur\'an problems ask for the maximum number of copies of a graph $H$ in an $n$-vertex, $F$-free graph, denoted by ex$(n,H,F)$. We show how to extend the new, localized approach of Brada\v{c}, Malec, and Tompkins to generalized Turán problems. We weight the copies of $H$ (typically taking $H=K_t$), instead of the edges, based on the size of the largest clique, path, or star containing the vertices of the copy of $H$, and in each case prove a tight upper bound on the sum of the weights. A consequence of our new localized theorems is an asymptotic determination of ex$(n,H,K_{1,r})$ for every $H$ having at least one dominating vertex and mex$(m,H,K_{1,r})$ for every $H$ having at least two dominating vertices.

Let H be a graph. We show that if r is large enough as a function of H , then the r -partite Turán graph maximizes the number of copies of H among all K r + 1-free graphs on a given number of vertices. This confirms a conjecture of Gerbner and Palmer.

We introduce a variant of the Localization game in which the cops only have visibility one, along with the corresponding optimization parameter, the one-visibility localization number ζ1. By developing lower bounds using isoperimetric inequalities, we give upper and lower bounds for ζ1 on k-ary trees with k≥2 that differ by a multiplicative constant, showing that the parameter is unbounded on k-ary trees. We provide a O(n) bound for Kh-minor free graphs of order n, and we show Cartesian grids meet this bound by determining their one-visibility localization number up to four values. We present upper bounds on ζ1 using pathwidth and the domination number and give upper bounds on trees via their depth and order. We conclude with open problems.

We study the Localization game on locally finite graphs and trees, where each vertex has finite degree. As in finite graphs, we prove that any locally finite graph contains a subdivision where one cop can capture the robber. In contrast to the finite case, for $n$ a positive integer, we construct a locally finite tree with localization number $n$ for any choice of $n$. Such trees contain uncountably many ends, and we show this is necessary by proving that graphs with countably many ends have localization number at most 2. We finish with questions on characterizing the localization number of locally finite trees.

The chromatic number of the random graph $\mathcal{G}(n,p)$ has long been studied and has inspired several landmark results. In the case where $p = d/n$, Achlioptas and Naor showed the chromatic number is asymptotically concentrated at $k_d$ or $k_d+1$, where $k_d$ is the smallest integer such that $d<2k_d\log k_d$. Kemkes et al. later proved the same result holds for $\mathcal{G}(n,d)$, the random $d$-regular graph. We consider the oriented chromatic number of the directed models $\vec{\mathcal{G}}(n,p)$ and $\vec{\mathcal{G}}(n,d)$, improving the best known upper bound from $O(d^2 2^d)$ to $O(\sqrt{e}^d)$.

The classical extremal problem is that of computing the maximum number of edges in an F -free graph. In particular, Turán’s theorem entirely resolves the case where F=K_r+1 . Later results, known as supersaturation theorems, proved that in a graph containing more edges than the extremal number, there must also be many copies of K_r+1 . Alon and Shikhelman introduced a broader class of extremal problems, asking for the maximum number of copies of a graph T in an F -free graph (so that T=K_2 is the classical extremal number). In this paper we determine some of these generalized extremal numbers when T and F are stars or cliques and prove some supersaturation results for them.

An n-lift of a graph G is a graph from which there is an n-to-1 covering map onto G. Amit, Linial, and Matoušek (2002) raised the question of whether the chromatic number of a random n-lift of K5 is concentrated on a single value. We consider this problem for G = Kd+1, and show that for fixed d ≥ 3 the chromatic number of a random lift of Kd is (asymptotically almost surely) either k or k + 1, where k is the smallest integer satisfying d < 2k log k. Moreover, we show that, for roughly half of the values of d, the chromatic number is concentrated on k. The argument for the upper-bound on the chromatic number uses the small subgraph conditioning method, and it can be extended to random n-lifts of G, for any fixed d-regular graph G.

The generalized Turán problem ex$(n,T,F)$ is to determine the maximal number of copies of a graph $T$ that can exist in an $F$-free graph on $n$ vertices. Recently, Gerbner and Palmer noted that the solution to the generalized Turán problem is often the original Turán graph. They gave the name "$F$-Turán-good" to graphs $T$ for which, for large enough $n$, the solution to the generalized Turán problem is realized by a Turán graph. They prove that the path graph on two edges, $P_2$, is $K_{r+1}$-Turán-good for all $r \ge 3$, but they conjecture that the same result should hold for all $P_\ell$. In this paper, using arguments based in flag algebras, we prove that the path on three edges, $P_3$, is also $K_{r+1}$-Turán-good for all $r \ge 3$.