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Let $d\ge 3$ be a fixed integer. Let $y:= y(p)$ be the probability that the root of an infinite $d$-regular tree belongs to an infinite cluster after $p$-bond-percolation. We show that for every constants $b,\alpha>0$ and $1<\lambda< d-1$, there exist constants $c,C>0$ such that the following holds. Let $G$ be a $d$-regular graph on $n$ vertices, satisfying that for every $U\subseteq V(G)$ with $|U|\le \frac{n}{2}$, $e(U,U^c)\ge b|U|$ and for every $U\subseteq V(G)$ with $|U|\le \log^Cn$, $e(U)\le (1+c)|U|$. Let $p=\frac{\lambda}{d-1}$. Then, with probability tending to one as $n$ tends to infinity, the largest component $L_1$ in the random subgraph $G_p$ of $G$ satisfies $\left|1-\frac{|L_1|}{yn}\right|\le \alpha$, and all the other components in $G_p$ are of order $O\left(\frac{\lambda\log n}{(\lambda-1)^2}\right)$. This generalises (and improves upon) results for random $d$-regular graphs.

We provide sufficient conditions for a regular graph $G$ of growing degree $d$, guaranteeing a phase transition in its random subgraph $G_p$ similar to that of $G(n,p)$ when $p\cdot d\approx 1$. These conditions capture several well-studied graphs, such as (percolation on) the complete graph $K_n$, the binary hypercube $Q^d$, $d$-regular expanders, and random $d$-regular graphs. In particular, this serves as a unified proof for these (and other) cases. Suppose that $G$ is a $d$-regular graph on $n$ vertices, with $d=\omega(1)$. Let $\epsilon>0$ be a small constant, and let $p=\frac{1+\epsilon}{d}$. Let $y(\epsilon)$ be the survival probability of a Galton-Watson tree with offspring distribution Po$(1+\epsilon)$. We show that if $G$ satisfies a (very) mild edge expansion requirement, and if one has fairly good control on the expansion of small sets in $G$, then typically the percolated random subgraph $G_p$ contains a unique giant component of asymptotic order $y(\epsilon)n$, and all the other components in $G_p$ are of order $O(\log n/\epsilon^2)$. We also show that this result is tight, in the sense that if one asks for a slightly weaker control on the expansion of small sets in $G$, then there are $d$-regular graphs $G$ on $n$ vertices, where typically the second largest component is of order $\Omega(d\log (n/d))=\omega(\log n)$.

Given a graph $G$ and $p\in [0,1]$, the random subgraph $G_p$ is obtained by retaining each edge of $G$ independently with probability $p$. We show that for every $\epsilon>0$, there exists a constant $C>0$ such that the following holds. Let $d\ge C$ be an integer, let $G$ be a $d$-regular graph and let $p\ge \frac{C}{d}$. Then, with probability tending to one as $|V(G)|$ tends to infinity, there exists a matching in $G_p$ covering at least $(1-\epsilon)|V(G)|$ vertices. We further show that for a wide family of $d$-regular graphs $G$, which includes the $d$-dimensional hypercube, for any $p\ge \frac{\log^5d}{d}$ with probability tending to one as $d$ tends to infinity, $G_p$ contains an induced subgraph on at least $(1-o(1))|V(G)|$ vertices, whose degrees are tightly concentrated around the expected average degree $dp$., Comment: 7 pages

Given a graph $G$ and probability $p$, we form the random subgraph $G_p$ by retaining each edge of $G$ independently with probability $p$. Given $d\in\mathbb{N}$ and constants $00$, we show that if every subset $S\subseteq V(G)$ of size exactly $\frac{c|V(G)|}{d}$ satisfies $|N(S)|\ge d|S|$ and $p=\frac{1+\varepsilon}{d}$, then the probability that $G_p$ does not contain a cycle of length $\Omega(\varepsilon^2c^2|V(G)|)$ is exponentially small in $|V(G)|$. As an intermediate step, we also show that given $k,d\in \mathbb{N}$ and a constant $\varepsilon>0$, if every subset $S\subseteq V(G)$ of size exactly $k$ satisfies $|N(S)|\ge kd$ and $p=\frac{1+\varepsilon}{d}$, then the probability that $G_p$ does not contain a path of length $\Omega(\varepsilon^2 kd)$ is exponentially small. We further discuss applications of these results to $K_{s,t}$-free graphs of maximal density., Comment: 7 pages

We estimate the minimum number of distance queries that is sufficient to reconstruct the binomial random graph $G(n,p)$ with constant diameter with high probability. We get a tight (up to a constant factor) answer for all $p>n^{-1+o(1)}$ outside "threshold windows" around $n^{-k/(k+1)+o(1)}$, $k\in\mathbb{Z}_{>0}$: with high probability the query complexity equals $\Theta(n^{4-d}p^{2-d})$, where $d$ is the diameter of the random graph. This demonstrates the following non-monotone behaviour: the query complexity jumps down at moments when the diameter gets larger; yet, between these moments the query complexity grows. We also show that there exists a non-adaptive algorithm that reconstructs the random graph with $O(n^{4-d}p^{2-d}\ln n)$ distance queries with high probability, and this is best possible.

We present a short and self-contained proof of a classical result due to Bollob\'as (1990): in the random hypercube process, with high probability the hitting time of connectedness equals the hitting time of having minimum degree at least one., Comment: 4 pages

Given positive integers $k \leq m$ and a graph $G$, a family of lists $L = \{L(v) : v \in V(G)\}$ is said to be a random $(k,m)$-list-assignment if for every $v \in V(G)$ the list $L(v)$ is a subset of $\{1, \ldots, m\}$ of size $k$, chosen uniformly at random and independently of the choices of all other vertices. An $n$-vertex graph $G$ is said to be a.a.s. $(k,m)$-colourable if $\lim_{n \to \infty} \mathbb{P}(G \textrm{ is } L-colourable) = 1$, where $L$ is a random $(k,m)$-list-assignment. We prove that if $m \gg n^{1/k^2} \Delta^{1/k}$ and $m \geq 3 k^2 \Delta$, where $\Delta$ is the maximum degree of $G$ and $k \geq 3$ is an integer, then $G$ is a.a.s. $(k,m)$-colourable. This is not far from being best possible, forms a continuation of the so-called palette sparsification results, and proves in a strong sense a conjecture of Casselgren. Additionally, we consider this problem under the additional assumption that $G$ is $H$-free for some graph $H$. For various graphs $H$, we estimate the smallest $m$ for which an $H$-free $n$-vertex graph $G$ is a.a.s. $(k,m)$-colourable. This extends and improves several results of Casselgren.

We study several basic problems about colouring the $p$-random subgraph $G_p$ of an arbitrary graph $G$, focusing primarily on the chromatic number and colouring number of $G_p$. In particular, we show that there exist infinitely many $k$-regular graphs $G$ for which the colouring number (i.e., degeneracy) of $G_{1/2}$ is at most $k/3 + o(k)$ with high probability, thus disproving the natural prediction that such random graphs must have colouring number at least $k/2 - o(k)$.

Let $Q^d$ be the $d$-dimensional binary hypercube. We say that $P=\{v_1,\ldots, v_k\}$ is an increasing path of length $k-1$ in $Q^d$, if for every $i\in [k-1]$ the edge $v_iv_{i+1}$ is obtained by switching some zero coordinate in $v_i$ to a one coordinate in $v_{i+1}$. Form a random subgraph $Q^d_p$ by retaining each edge in $E(Q^d)$ independently with probability $p$. We show that there is a phase transition with respect to the length of a longest increasing path around $p=\frac{e}{d}$. Let $\alpha$ be a constant and let $p=\frac{\alpha}{d}$. When $\alphae$, whp there is a path of length $d-2$ in $Q^d_p$, and in fact, whether it is of length $d-2, d-1$, or $d$ depends on whether the all-zero and all-one vertices percolate or not.

A graph is called $d$-rigid if there exists a generic embedding of its vertex set into $\mathbb{R}^d$ such that every continuous motion of the vertices that preserves the lengths of all edges actually preserves the distances between all pairs of vertices. The rigidity of a graph is the maximal $d$ such that the graph is $d$-rigid. We present new sufficient conditions for the $d$-rigidity of a graph in terms of the existence of ``rigid partitions'' -- partitions of the graph that satisfy certain connectivity properties. This extends previous results by Crapo, Lindemann, and Lew, Nevo, Peled and Raz. As an application, we present new results on the rigidity of highly-connected graphs, random graphs, random bipartite graphs, pseudorandom graphs, and dense graphs. In particular, we prove that random $C d\log d$-regular graphs are typically $d$-rigid, demonstrate the existence of a giant $d$-rigid component in sparse random binomial graphs, and show that the rigidity of relatively sparse random binomial bipartite graphs is roughly the same as that of the complete bipartite graph, which we consider an interesting phenomenon. Furthermore, we show that a graph admitting $\binom{d+1}{2}$ disjoint connected dominating sets is $d$-rigid. This implies a weak version of the Lov\'asz--Yemini conjecture on the rigidity of highly-connected graphs. We also present an alternative short proof for a recent result by Lew, Nevo, Peled, and Raz, which asserts that the hitting time for $d$-rigidity in the random graph process typically coincides with the hitting time for minimum degree $d$., Comment: 30 pages. In this updated version, we have added a theorem concerning the rigidity of dense graphs and incorporated references to Vill\'anyi's recent resolution of the Lov\'asz-Yemini conjecture

We present a relatively short and self-contained proof of the classical result on component sizes in the supercritical percolation on the high dimensional binary cube, due to Ajtai, Koml\'os and Szemer\'edi (1982) and to Bollob\'as, Kohayakawa and \L uczak (1992).

A classical problem, due to Gerencs\'er and Gy\'arf\'as from 1967, asks how large a monochromatic connected component can we guarantee in any $r$-edge colouring of $K_n$? We consider how big a connected component can we guarantee in any $r$-edge colouring of $K_n$ if we allow ourselves to use up to $s$ colours. This is actually an instance of a more general question of Bollob\'as from about 20 years ago which asks for a $k$-connected subgraph in the same setting. We complete the picture in terms of the approximate behaviour of the answer by determining it up to a logarithmic term, provided $n$ is large enough. We obtain more precise results for certain regimes which solve a problem of Liu, Morris and Prince from 2007, as well as disprove a conjecture they pose in a strong form. We also consider a generalisation in a similar direction of a question first considered by Erd\H{o}s and R\'enyi in 1956, who considered given $n$ and $m$, what is the smallest number of $m$-cliques which can cover all edges of $K_n$? This problem is essentially equivalent to the question of what is the minimum number of vertices that are certain to be incident to at least one edge of some colour in any $r$-edge colouring of $K_n$. We consider what happens if we allow ourselves to use up to $s$ colours. We obtain a more complete understanding of the answer to this question for large $n$, in particular determining it up to a constant factor for all $1\le s \le r$, as well as obtaining much more precise results for various ranges including the correct asymptotics for essentially the whole range.

We provide a sufficient condition on the isoperimetric properties of a regular graph $G$ of growing degree $d$, under which the random subgraph $G_p$ typically undergoes a phase transition around $p=\frac{1}{d}$ which resembles the emergence of a giant component in the binomial random graph model $G(n,p)$. We further show that this condition is tight. More precisely, let $d=\omega(1)$, let $\epsilon>0$ be a small enough constant, and let $p \cdot d=1+\epsilon$. We show that if $C$ is sufficiently large and $G$ is a $d$-regular $n$-vertex graph where every subset $S\subseteq V(G)$ of order at most $\frac{n}{2}$ has edge-boundary of size at least $C|S|$, then $G_p$ typically has a unique linear sized component, whose order is asymptotically $y(\epsilon)n$, where $y(\epsilon)$ is the survival probability of a Galton-Watson tree with offspring distribution Po$(1+\epsilon)$. We further give examples to show that this result is tight both in terms of its dependence on $C$, and with respect to the order of the second-largest component. We also consider a more general setting, where we only control the expansion of sets up to size $k$. In this case, we show that if $G$ is such that every subset $S\subseteq V(G)$ of order at most $k$ has edge-boundary of size at least $d|S|$ and $p$ is such that $p\cdot d \geq 1 + \epsilon$, then $G_p$ typically contains a component of order $\Omega(k)$.

We find asymptotics of the maximum size of a chordal subgraph in a binomial random graph $G(n,p)$, for $p=\mathrm{const}$ and $p=n^{-\alpha+o(1)}$.

It is known that the complete graph $K_n$ contains a pancyclic subgraph with $n+(1+o(1))\cdot \log _2 n$ edges, and that there is no pancyclic graph on $n$ vertices with fewer than $n+\log _2 (n-1) -1$ edges. We show that, with high probability, $G(n,p)$ contains a pancyclic subgraph with $n+(1+o(1))\log_2 n$ edges for $p \ge p^*$, where $p^*=(1+o(1))\ln n/n$, right above the threshold for pancyclicity.

A crown with $k$ spikes is an edge-disjoint union of a cycle $C$ and a matching $M$ of size $k$ such that each edge of $M$ has exactly one vertex in common with $C$. We prove that if $G$ is an $(n,d,\lambda)$-graph with $\lambda/d\le 0.001$ and $d$ is large enough, then $G$ contains a crown on $n$ vertices with $\lfloor n/2\rfloor$ spikes. As a consequence, such $G$ contains a Hamilton cycle in its square $G^2$.

It is known that many different types of finite random subgraph models undergo quantitatively similar phase transitions around their percolation thresholds, and the proofs of these results rely on isoperimetric properties of the underlying host graph. Recently, the authors showed that such a phase transition occurs in a large class of regular high-dimensional product graphs, generalising a classic result for the hypercube. In this paper we give new isoperimetric inequalities for such regular high-dimensional product graphs, which generalise the well-known isoperimetric inequality of Harper for the hypercube, and are asymptotically sharp for a wide range of set sizes. We then use these isoperimetric properties to investigate the structure of the giant component $L_1$ in supercritical percolation on these product graphs, that is, when $p=\frac{1+\epsilon}{d}$, where $d$ is the degree of the product graph and $\epsilon>0$ is a small enough constant. We show that typically $L_1$ has edge-expansion $\Omega\left(\frac{1}{d\ln d}\right)$. Furthermore, we show that $L_1$ likely contains a linear-sized subgraph with vertex-expansion $\Omega\left(\frac{1}{d\ln d}\right)$. These results are best possible up to the logarithmic factor in $d$. Using these likely expansion properties, we determine, up to small polylogarithmic factors in $d$, the likely diameter of $L_1$ as well as the typical mixing time of a lazy random walk on $L_1$. Furthermore, we show the likely existence of a path of length $\Omega\left(\frac{n}{d\ln d}\right)$. These results not only generalise, but also improve substantially upon the known bounds in the case of the hypercube, where in particular the likely diameter and typical mixing time of $L_1$ were previously only known to be polynomial in $d$.

Extremal properties of sparse graphs, randomly perturbed by the binomial random graph are considered. It is known that every $n$-vertex graph $G$ contains a complete minor of order $\Omega(n/\alpha(G))$. We prove that adding $\xi n$ random edges, where $\xi > 0$ is arbitrarily small yet fixed, to an $n$-vertex graph $G$ satisfying $\alpha(G) \leq \zeta(\xi)n$ asymptotically almost surely results in a graph containing a complete minor of order $\tilde \Omega \left( n/\sqrt{\alpha(G)}\right)$; this result is tight up to the implicit logarithmic terms. For complete topological minors, we prove that there exists a constant $C>0$ such that adding $C n$ random edges to a graph $G$ satisfying $\delta(G) = \omega(1)$, asymptotically almost surely results in a graph containing a complete topological minor of order $\tilde \Omega(\min\{\delta(G),\sqrt{n}\})$; this result is tight up to the implicit logarithmic terms. Finally, extending results of Bohman, Frieze, Krivelevich, and Martin for the dense case, we analyse the asymptotic behaviour of the vertex-connectivity and the diameter of randomly perturbed sparse graphs.

A classical vertex Ramsey result due to Ne\v{s}et\v{r}il and R\"odl states that given a finite family of graphs $\mathcal{F}$, a graph $A$ and a positive integer $r$, if every graph $B\in\mathcal{F}$ has a $2$-vertex-connected subgraph which is not a subgraph of $A$, then there exists an $\mathcal{F}$-free graph which is vertex $r$-Ramsey with respect to $A$. We prove that this sufficient condition for the existence of an $\mathcal{F}$-free graph which is vertex $r$-Ramsey with respect to $A$ is also necessary for large enough number of colours $r$. We further show a generalisation of the result to a family of graphs and the typical existence of such a subgraph in a dense binomial random graph., Comment: One figure

We consider bond percolation on high-dimensional product graphs $G=\square_{i=1}^tG^{(i)}$, where $\square$ denotes the Cartesian product. We call the $G^{(i)}$ the base graphs and the product graph $G$ the host graph. Very recently, Lichev showed that, under a mild requirement on the isoperimetric properties of the base graphs, the component structure of the percolated graph $G_p$ undergoes a phase transition when $p$ is around $\frac{1}{d}$, where $d$ is the average degree of the host graph. In the supercritical regime, we strengthen Lichev's result by showing that the giant component is in fact unique, with all other components of order $o(|G|)$, and determining the sharp asymptotic order of the giant. Furthermore, we answer two questions posed by Lichev: firstly, we provide a construction showing that the requirement of bounded-degree is necessary for the likely emergence of a linear order component; secondly, we show that the isoperimetric requirement on the base graphs can be, in fact, super-exponentially small in the dimension. Finally, in the subcritical regime, we give an example showing that in the case of irregular high-dimensional product graphs, there can be a polynomially large component with high probability, very much unlike the quantitative behaviour seen in the Erd\H{o}s-R\'enyi random graph and in the percolated hypercube, and in fact in any regular high-dimensional product graphs, as shown by the authors in a companion paper.

We prove that for every non-trivial hereditary family of graphs ${\cal P}$ and for every fixed $p \in (0,1)$, the maximum possible number of edges in a subgraph of the random graph $G(n,p)$ which belongs to ${\cal P}$ is, with high probability, $$ \left(1-\frac{1}{k-1}+o(1)\right)p{n \choose 2}, $$ where $k$ is the minimum chromatic number of a graph that does not belong to ${\cal P}$.

We prove that for every $\varepsilon > 0$ there is $c_0$ such that if $G\sim G(n,c/n)$, $c\ge c_0$, then with high probability $G$ can be covered by at most $(1+\varepsilon)\cdot \frac{1}{2}ce^{-c} \cdot n$ vertex disjoint paths, which is essentially tight. This is equivalent to showing that, with high probability, at most $(1+\varepsilon)\cdot \frac{1}{2}ce^{-c} \cdot n$ edges can be added to $G$ to create a Hamiltonian graph.

We consider percolation on high-dimensional product graphs, where the base graphs are regular and of bounded order. In the subcritical regime, we show that typically the largest component is of order logarithmic in the number of vertices. In the supercritical regime, our main result recovers the sharp asymptotic of the order of the largest component, and shows that all the other components are typically of order logarithmic in the number of vertices. In particular, we show that this phase transition is quantitatively similar to the one of the binomial random graph. This generalises the results of Ajtai, Koml\'os, and Szemer\'edi and of Bollob\'as, Kohayakawa, and \L{}uczak who showed that the $d$-dimensional hypercube, which is the $d$-fold Cartesian product of an edge, undergoes a phase transition quantitatively similar to the one of the binomial random graph.

We introduce a model of a controlled random graph process. In this model, the edges of the complete graph $K_n$ are ordered randomly and then revealed, one by one, to a player called Builder. He must decide, immediately and irrevocably, whether to purchase each observed edge. The observation time is bounded by parameter $t$, and the total budget of purchased edges is bounded by parameter $b$. Builder's goal is to devise a strategy that, with high probability, allows him to construct a graph of purchased edges possessing a target graph property $\mathcal{P}$, all within the limitations of observation time and total budget. We show the following: (a) Builder has a strategy to achieve $k$-vertex-connectivity at the hitting time for this property by purchasing at most $c_kn$ edges for an explicit $c_k1$; this is optimal in the sense that $C$ cannot be arbitrarily close to $1$. This substantially extends the classical hitting time result for Hamiltonicity due to Ajtai--Koml\'os--Szemer\'edi and Bollob\'as. (c) Builder has a strategy to create a perfect matching by time $(1+\varepsilon)n\log{n}/2$ while purchasing at most $(1+\varepsilon)n/2$ edges (which is optimal). (d) Builder has a strategy to create a copy of a given $k$-vertex tree if $t\ge b\gg\max\{(n/t)^{k-2},1\}$, and this is optimal; (e) For $\ell=2k+1$ or $\ell=2k+2$, Builder has a strategy to create a copy of a cycle of length $\ell$ if $b\gg\max \{n^{k+2}/t^{k+1},n/\sqrt{t}\}$, and this is optimal., Comment: 30 pages, 2 figures; added a stronger statement for k-connectivity with a simpler and shorter proof

Let $G$ be an $n$-vertex graph, where $\delta(G) \geq \delta n$ for some $\delta := \delta(n)$. A result of Bohman, Frieze and Martin from 2003 asserts that if $\alpha(G) = O \left(\delta^2 n \right)$, then perturbing $G$ via the addition of $\omega \left(\frac{\log(1/\delta)}{\delta^3} \right)$ random edges, asymptotically almost surely (a.a.s. hereafter) results in a Hamiltonian graph. This bound on the size of the random perturbation is only tight when $\delta$ is independent of $n$ and deteriorates as to become uninformative when $\delta = \Omega \left(n^{-1/3} \right)$. We prove several improvements and extensions of the aforementioned result. First, keeping the bound on $\alpha(G)$ as above and allowing for $\delta = \Omega(n^{-1/3})$, we determine the correct order of magnitude of the number of random edges whose addition to $G$ a.a.s. results in a pancyclic graph. Our second result ventures into significantly sparser graphs $G$; it delivers an almost tight bound on the size of the random perturbation required to ensure pancyclicity a.a.s., assuming $\delta(G) = \Omega \left((\alpha(G) \log n)^2 \right)$ and $\alpha(G) \delta(G) = O(n)$. Assuming the correctness of Chv\'atal's toughness conjecture, allows for the mitigation of the condition $\alpha(G) = O \left(\delta^2 n \right)$ imposed above, by requiring $\alpha(G) = O(\delta(G))$ instead; our third result determines, for a wide range of values of $\delta(G)$, the correct order of magnitude of the size of the random perturbation required to ensure the a.a.s. pancyclicity of $G$. For the emergence of nearly spanning cycles, our fourth result determines, under milder conditions, the correct order of magnitude of the size of the random perturbation required to ensure that a.a.s. $G$ contains such a cycle., Comment: 21 pages, 4 figures

We present a short and self-contained proof of the choosability version of Brooks' theorem., Comment: 2 pages

In 2004, Frieze, Krivelevich and Martin [17] established the emergence of a giant component in random subgraphs of pseudo-random graphs. We study several typical properties of the giant component, most notably its expansion characteristics. We establish an asymptotic vertex expansion of connected sets in the giant by a factor of $\tilde{O}\left(\epsilon^2\right)$. From these expansion properties, we derive that the diameter of the giant is typically $O_{\epsilon}\left(\log n\right)$, and that the mixing time of a lazy random walk on the giant is asymptotically $O_{\epsilon}\left(\log^2 n\right)$. We also show similar asymptotic expansion properties of (not necessarily connected) linear sized subsets in the giant, and the typical existence of a large expander as a subgraph.

We consider supercritical site percolation on the $d$-dimensional hypercube $Q^d$. We show that typically all components in the percolated hypercube, besides the giant, are of size $O(d)$. This resolves a conjecture of Bollob\'as, Kohayakawa, and {\L}uczak from 1994., Comment: 6 pages

We propose the following conjecture extending Dirac's theorem: if $G$ is a graph with $n\ge 3$ vertices and minimum degree $\delta(G)\ge n/2$, then in every orientation of $G$ there is a Hamilton cycle with at least $\delta(G)$ edges oriented in the same direction. We prove an approximate version of this conjecture, showing that minimum degree $n/2 + O(k)$ guarantees a Hamilton cycle with at least $(n+k)/2$ edges oriented in the same direction. We also study the analogous problem for random graphs, showing that if the edge probability $p = p(n)$ is above the Hamiltonicity threshold, then, with high probability, in every orientation of $G \sim G(n,p)$ there is a Hamilton cycle with $(1-o(1))n$ edges oriented in the same direction.

We provide a short and self-contained proof of the classical result of Kostochka and of Thomason, ensuring that every graph of average degree $d$ has a complete minor of order $d/\sqrt{\log d}$., Comment: 3 pages; revised version, minor changes

We consider the performance of the Depth First Search (DFS) algorithm on the random graph $G\left(n,\frac{1+\epsilon}{n}\right)$, $\epsilon>0$ a small constant. Recently, Enriquez, Faraud and M\'enard [2] proved that the stack $U$ of the DFS follows a specific scaling limit, reaching the maximal height of $(1+o_{\epsilon}(1))\epsilon^2n$. Here we provide a simple analysis for the typical length of a maximum path discovered by the DFS., Comment: minor changes

It is well-known that the behaviour of a random subgraph of a $d$-dimensional hypercube, where we include each edge independently with probability $p$, undergoes a phase transition when $p$ is around $\frac{1}{d}$. More precisely, standard arguments show that just below this value of $p$ all components of this graph have order $O(d)$ with probability tending to one as $d \to \infty$ (whp for short), whereas Ajtai, Koml\'{o}s and Szemer\'{e}di [Largest random component of a $k$-cube, Combinatorica 2 (1982), no. 1, 1--7; MR0671140] showed that just above this value, in the supercritical regime, whp there is a unique `giant' component of order $\Theta\left(2^d\right)$. We show that whp the vertex-expansion of the giant component is inverse polynomial in $d$. As a consequence we obtain polynomial in $d$ bounds on the diameter of the giant component and the mixing time of the lazy random walk on the giant component, answering questions of Bollob\'{a}s, Kohayakawa and {\L}uczak [On the diameter and radius of random subgraphs of the cube, Random Structures and Algorithms 5 (1994), no. 5, 627--648; MR1300592] and of Pete [A note on percolation on $\mathbb{Z}^d$: isoperimetric profile via exponential cluster repulsion, Electron. Commun. Probab. 13 (2008), 377--392; MR2415145]. Furthermore, our results imply lower bounds on the circumference and Hadwiger number of a random subgraph of the hypercube in this regime of $p$ which are tight up to polynomial factors in $d$., Comment: 29 pages, this work reuses parts from an earlier work of the same authors (arXiv:2106.04249, not to be published), we strengthen the main result of that work and give further applications

We consider vertex percolation on pseudo-random $d-$regular graphs. The previous study by the second author established the existence of phase transition from small components to a linear (in $\frac{n}{d}$) sized component, at $p=\frac{1}{d}$. In the supercritical regime, our main result recovers the sharp asymptotic of the size of the largest component, and shows that all other components are typically much smaller. Furthermore, we consider other typical properties of the largest component such as the number of edges, existence of a long cycle and expansion. In the subcritical regime, we strengthen the upper bound on the likely component size.

We show that with high probability the random graph $G_{n, 1/2}$ has an induced subgraph of linear size, all of whose degrees are congruent to $r\pmod q$ for any fixed $r$ and $q\geq 2$. More generally, the same is true for any fixed distribution of degrees modulo $q$. Finally, we show that with high probability we can partition the vertices of $G_{n, 1/2}$ into $q+1$ parts of nearly equal size, each of which induces a subgraph all of whose degrees are congruent to $r\pmod q$. Our results resolve affirmatively a conjecture of Scott, who addressed the case $q=2$.

We present a modification of the Depth first search algorithm, suited for finding long induced paths. We use it to give simple proofs of the following results. We show that the induced size-Ramsey number of paths satisfies $\hat{R}_{\mathrm{ind}}(P_n)\leq 5\cdot 10^7n$, thus giving an explicit constant in the linear bound, improving the previous bound with a large constant from a regularity lemma argument by Haxell, Kohayakawa and {\L}uczak. We also provide a bound for the $k$-color version, showing that $\hat{R}_{\mathrm{ind}}^k(P_n)=O(k^3\log^4k)n$. Finally, we present a new short proof of the fact that the binomial random graph in the supercritical regime, $G(n,\frac{1+\varepsilon}{n})$, contains typically an induced path of length $\Theta(\varepsilon^2) n$., Comment: to appear in CPC

Analogous to the case of the binomial random graph $G(d+1,p)$, it is known that the behaviour of a random subgraph of a $d$-dimensional hypercube, where we include each edge independently with probability $p$, which we denote by $Q^d_p$, undergoes a phase transition around the critical value of $p=\frac{1}{d}$. More precisely, standard arguments show that significantly below this value of $p$, with probability tending to one as $d \to \infty$ (whp for short) all components of this graph have order $O(d)$, whereas Ajtai, Koml\'{o}s and Szemer\'{e}di showed that significantly above this value, in the \emph{supercritical regime}, whp there is a unique `giant' component of order $\Theta\left(2^d\right)$. In $G(d+1,p)$ much more is known about the complex structure of the random graph which emerges in this supercritical regime. For example, it is known that in this regime whp $G(d+1,p)$ contains paths and cycles of length $\Omega(d)$, as well as complete minors of order $\Omega\left(\sqrt{d}\right)$. In this paper we obtain analogous results in $Q^d_p$. In particular, we show that for supercritical $p$, i.e., when $p=\frac{1+\epsilon}{d}$ for a positive constant $\epsilon$, whp $Q^d_p$ contains a cycle of length $\Omega\left(\frac{2^d}{d^3(\log d)^3} \right)$ and a complete minor of order $\Omega\left(\frac{2^{\frac{d}{2}}}{d^3(\log d)^3 }\right)$. In order to prove these results, we show that whp the largest component of $Q^d_p$ has good edge-expansion properties, a result of independent interest. We also consider the genus of $Q^d_p$ and show that, in this regime of $p$, whp the genus is $\Omega\left(2^d\right)$., Comment: 20 pages, the results of this paper are superseded by those in arXiv:2111.06752 and this paper will not be published

We show that for any $d=d(n)$ with $d_0(\epsilon) \le d =o(n)$, with high probability, the size of a largest induced cycle in the random graph $G(n,d/n)$ is $(2\pm \epsilon)\frac{n}{d}\log d$. This settles a long-standing open problem in random graph theory., Comment: to appear in RSA. arXiv admin note: substantial text overlap with arXiv:2102.09289

In the $\left(1:b\right)$ component game played on a graph $G$, two players, Maker and Breaker, alternately claim~$1$ and~$b$ previously unclaimed edges of $G$, respectively. Maker's aim is to maximise the size of a largest connected component in her graph, while Breaker is trying to minimise it. We show that the outcome of the game on the binomial random graph is strongly correlated with the appearance of a nonempty $(b+2)$-core in the graph. For any integer $k$, the $k$-core of a graph is its largest subgraph of minimum degree at least $k$. Pittel, Spencer and Wormald showed in 1996 that for any $k\ge3$ there exists an explicitly defined constant $c_{k}$ such that $p=c_{k}/n$ is the threshold function for the appearance of the $k$-core in $G(n,p)$. More precisely, $G(n,c/n)$ has WHP a linear-size $k$-core when the constant $c>c_{k}$, and an empty $k$-core when $cc_{b+2}$, while Breaker can WHP prevent Maker from building larger than polylogarithmic-size components if $c

We study the multicolour discrepancy of spanning trees and Hamilton cycles in graphs. As our main result, we show that under very mild conditions, the $r$-colour spanning-tree discrepancy of a graph $G$ is equal, up to a constant, to the minimum $s$ such that $G$ can be separated into $r$ equal parts by deleting $s$ vertices. This result arguably resolves the question of estimating the spanning-tree discrepancy in essentially all graphs of interest. In particular, it allows us to immediately deduce as corollaries most of the results that appear in a recent paper of Balogh, Csaba, Jing and Pluh\'{a}r, proving them in wider generality and for any number of colours. We also obtain several new results, such as determining the spanning-tree discrepancy of the hypercube. For the special case of graphs possessing certain expansion properties, we obtain exact asymptotic bounds. We also study the multicolour discrepancy of Hamilton cycles in graphs of large minimum degree, showing that in any $r$-colouring of the edges of a graph with $n$ vertices and minimum degree at least $\frac{r+1}{2r}n + d$, there must exist a Hamilton cycle with at least $\frac{n}{r} + 2d$ edges in some colour. This extends a result of Balogh et al., who established the case $r = 2$. The constant $\frac{r+1}{2r}$ in this result is optimal; it cannot be replaced by any smaller constant., Comment: 25 pages, 5 figures

We prove that for every graph $H$ of maximum degree at most $3$ and for every positive integer $q$ there is a finite $f=f(H,q)$ such that every $K_f$-minor contains a subdivision of $H$ in which every edge is replaced by a path whose length is divisible by $q$. For the case of cycles we show that for $f=O(q \log q)$ every $K_f$-minor contains a cycle of length divisible by $q$, and observe that this settles a recent problem of Friedman and the second author about cycles in (weakly) expanding graphs., Comment: Revised version, minor changes

We present an explicit connected spanning structure that appears in a random graph just above the connectivity threshold with high probability., Comment: 16 pages

We prove that every graph $G$ on $n$ vertices with no isolated vertices contains an induced subgraph of size at least $n/10000$ with all degrees odd. This solves an old and well-known conjecture in graph theory.

We study the set ${\cal L}(G)$ of lengths of all cycles that appear in a random $d$-regular $G$ on $n$ vertices for a fixed $d\geq 3$, as well as in Erd\H{o}s--R\'enyi random graphs on $n$ vertices with a fixed average degree $c>1$. Fundamental results on the distribution of cycle counts in these models were established in the 1980's and early 1990's, with a focus on the extreme lengths: cycles of fixed length, and cycles of length linear in $n$. Here we derive, for a random $d$-regular graph, the limiting probability that ${\cal L}(G)$ simultaneously contains the entire range $\{\ell,\ldots,n\}$ for $\ell\geq 3$, as an explicit expression $\theta_\ell=\theta_\ell(d)\in(0,1)$ which goes to $1$ as $\ell\to\infty$. For the random graph ${\cal G}(n,p)$ with $p=c/n$, where $c\geq C_0$ for some absolute constant $C_0$, we show the analogous result for the range $\{\ell,\ldots,(1-o(1))L_{\max}(G)\}$, where $L_{\max}$ is the length of a longest cycle in $G$. The limiting probability for ${\cal G}(n,p)$ coincides with $\theta_\ell$ from the $d$-regular case when $c$ is the integer $d-1$. In addition, for the directed random graph ${\cal D}(n,p)$ we show results analogous to those on ${\cal G}(n,p)$, and for both models we find an interval of $c \epsilon^2 n$ consecutive cycle lengths in the slightly supercritical regime $p=\frac{1+\epsilon}n$., Comment: 11 pages, 3 figures

We prove that a random graph $G(n,p)$, with $p$ above the Hamiltonicity threshold, is typically such that for any $r$-colouring of its edges there exists a Hamilton cycle with at least $(2/(r+ 1)-o(1))n$ edges of the same colour. This estimate is asymptotically optimal., Comment: 20 pages, minor corrections

We develop a general embedding method based on the Friedman-Pippenger tree embedding technique (1987) and its algorithmic version, essentially due to Aggarwal et al. (1996), enhanced with a roll-back idea allowing to sequentially retrace previously performed embedding steps. We use this method to obtain the following results. -We show that the size-Ramsey number of logarithmically long subdivisions of bounded degree graphs is linear in their number of vertices, settling a conjecture of Pak (2002). -We give a deterministic, polynomial time online algorithm for finding vertex-disjoint paths of prescribed length between given pairs of vertices in an expander graph. Our result answers a question of Alon and Capalbo (2007). -We show that relatively weak bounds on the spectral ratio of $d$-regular graphs force the existence of a topological minor of $K_t$ where $t=(1-o(1))d$. We also exhibit a construction which shows that the theoretical maximum $t=d+1$ cannot be attained even if $\lambda=O(\sqrt{d})$. This answers a question of Fountoulakis, K\"uhn and Osthus (2009)., Comment: improved exposition; typos fixed; new section and results about topological minors

The $r$-size-Ramsey number $\hat{R}_r(H)$ of a graph $H$ is the smallest number of edges a graph $G$ can have, such that for every edge-coloring of $G$ with $r$ colors there exists a monochromatic copy of $H$ in $G$. For a graph $H$, we denote by $H^q$ the graph obtained from $H$ by subdividing its edges with $q{-}1$ vertices each. In a recent paper of Kohayakawa, Retter and R{\"o}dl, it is shown that for all constant integers $q,r\geq 2$ and every graph $H$ on $n$ vertices and of bounded maximum degree, the $r$-size-Ramsey number of $H^q$ is at most $(\log n)^{20(q-1)}n^{1+1/q}$, for $n$ large enough. We improve upon this result using a significantly shorter argument by showing that $\hat{R}_r(H^q)\leq O(n^{1+1/q})$ for any such graph $H$., Comment: 12 pages, 1 figure

Let $G$ be a graph of minimum degree at least $k$ and let $G_p$ be the random subgraph of $G$ obtained by keeping each edge independently with probability $p$. We are interested in the size of the largest complete minor that $G_p$ contains when $p = \frac{1+\varepsilon}{k}$ with $\varepsilon >0$. We show that with high probability $G_p$ contains a complete minor of order $\tilde{\Omega}(\sqrt{k})$, where the $\sim$ hides a polylogarithmic factor. Furthermore, in the case where the order of $G$ is also bounded above by a constant multiple of $k$, we show that this polylogarithmic term can be removed, giving a tight bound., Comment: 12 pages, small changes in exposition and a simplification of the proof of Lemma 5