Weak Saturation Numbers in Random Graphs

The H-free process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal H-free graph obtained at the end of the process. When H is strictly 2-balanced, we show that for some c>0, with high probability as n→∞, the minimum degree in G is at least \(cn^{1-(v_{H}-2)/(e_{H}-1)}(\log n)^{1/(e_{H}-1)}\). This gives new lower bounds for the Turán numbers of certain bipartite graphs, such as the complete bipartite graphs K r,r with r≥5. When H is a complete graph K s with s≥5 we show that for some C>0, with high probability the independence number of G is at most \(Cn^{2/(s+1)}(\log n)^{1-1/(e_{H}-1)}\). This gives new lower bounds for Ramsey numbers R(s,t) for fixed s≥5 and t large. We also obtain new bounds for the independence number of G for other graphs H, including the case when H is a cycle. Our proofs use the differential equations method for random graph processes to analyse the evolution of the process, and give further information about the structure of the graphs obtained, including asymptotic formulae for a broad class of subgraph extension variables.

We say that a graph $G$ is $K_r$-saturated if $G$ contains no copy of $K_r$ but adding any new edge to $G$ creates a copy of $K_r$. Let $sat(n,K_r,t)$ be the minimum number of edges in a $K_r$-saturated graph on $n$ vertices with minimum degree at least $t$. Day showed that for fixed $r \geq 3$ and $t \geq r-2$, $sat(n,K_r,t)=tn-c(r,t)$ for large enough $n$, where $c(r,t)$ is a constant depending on $r$ and $t$, and proved the bounds $$ 2^t t^{3/2} \ll_r c(r,t) \leq t^{t^{2t^2}} $$for fixed $r$ and large $t$. In this paper we show that for fixed $r$ and large $t$, the order of magnitude of $c(r,t)$ is given by $c(r,t)=\Theta_r \left(4^t t^{-1/2} \right)$. Moreover, we investigate the dependence on $r$, obtaining the estimates$$ \frac{4^{t-r}}{\sqrt{t-r+3}} + r^2 \ll c(r,t) \ll \frac{4^{t-r} \min{(r,\sqrt{t-r+3})}}{\sqrt{t-r+3}} + r^2 \ . $$We further show that for all $r$ and $t$, there is a finite collection of graphs such that all extremal graphs are blow-ups of graphs in the collection. Using similar ideas, we show that every large $K_r$-saturated graph with $e$ edges has a vertex cover of size $O(e / \log e)$, uniformly in $r \geq 3$. This strengthens a previous result of Pikhurko. We also provide examples for which this bound is tight. A key ingredient in the proofs is a new version of Bollobás's Two Families Theorem.

Expander properties and the cover time of random intersection graphs

ABSTRACTAn independent set of size in a finite undirected graph is a set of vertices of the graph, no two of which are connected by an edge. Let be the number of independent sets of size in the graph and let . In 1987, Alavi, Malde, Schwenk, and Erdős asked if the independent set sequence of a tree is unimodal (the sequence goes up and then down). This problem is still open. In 2006, Levit and Mandrescu showed that the last third of the independent set sequence of a tree is decreasing. We show that the first 46.8% of the independent set sequence of a random tree is increasing with (exponentially) high probability as the number of vertices goes to infinity. So, the question of Alavi, Malde, Schwenk, and Erdős is “four‐fifths true,” with high probability. We also show unimodality of the independent set sequence of Erdős–Rényi random graphs, when the expected degree of a single vertex is large (with [exponentially] high probability as the number of vertices in the graph goes to infinity, except for a small region near the mode). A weaker result is shown for random regular graphs. The structure of independent sets of size as varies is of interest in probability, statistical physics, combinatorics, and computer science.

Motivated by an application in controllability we consider maximum matchings in random bipartite graphs G = (A, B). First we analyse Karp–Sipser's algorithm to determine the asymptotic size of maximum matchings in random bipartite graphs with a fixed degree distribution. We then allow an adversary to delete one edge adjacent to every vertex in A in the more restricted model where each vertex in A chooses d neighbours uniformly at random from B.

We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n → ∞. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability all the components are small, and other conditions that imply that with high probability there is a giant component and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the results by Molloy and Reed on the size of the largest component in a random graph with a given degree sequence.We further obtain a new sharp result for the giant component just above the threshold, generalizing the case of G(n,p) with np = 1 + ω(n)n−1/3, where ω(n) → ∞ arbitrarily slowly.Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009

We consider a system of N servers inter-connected by some underlying graph topology G N . Tasks with unit-mean exponential processing times arrive at the various servers as independent Poisson processes of rate λ. Each incoming task is irrevocably assigned to whichever server has the smallest number of tasks among the one where it appears and its neighbors in G N . The above model arises in the context of load balancing in large-scale cloud networks and data centers, and has been extensively investigated in the case G N is a clique. Since the servers are exchangeable in that case, mean-field limits apply, and in particular it has been proved that for any λ < 1, the fraction of servers with two or more tasks vanishes in the limit as N → ∞. For an arbitrary graph G N , mean-field techniques break down, complicating the analysis, and the queue length process tends to be worse than for a clique. Accordingly, a graph G N is said to be N -optimal or √ N -optimal when the queue length process on G N is equivalent to that on a clique on an N -scale or √ N -scale, respectively. We prove that if G N is an Erdöo s-Rényi random graph with average degree d ( N ), then with high probability it is N -optimal and √ N -optimal if d ( N ) → ∞ and d ( N )/√ N log( N )) → ∞ as N → ∞, respectively. This demonstrates that optimality can be maintained at N -scale and √ N -scale while reducing the number of connections by nearly a factor N and √ N /log( N ) compared to a clique, provided the topology is suitably random. It is further shown that if G N contains Θ( N ) bounded-degree nodes, then it cannot be N -optimal. In addition, we establish that an arbitrary graph G N is N -optimal when its minimum degree is N - o ( N ), and may not be N -optimal even when its minimum degree is cN + o ( N ) for any 0 < c < 1/2. Simulation experiments are conducted for various scenarios to corroborate the asymptotic results.

Following this advice, we should look over the big questions, then doodle and sketch out some approaches. If you are an expert, then this is easy to do, but most people do not want to wait to become an expert before looking at interesting problems. Graph theory, our area of expertise, has many hard-to-solve questions. Some hark back to the recreational roots of the area yet still keep their mystery. These ‘‘acorns’’ can be planted on the backs of envelopes, on a blackboard, and over a coffee. Our goal is to collect some of these conjectures—arguably some of the most intriguing—in one place. We present ten conjectures in graph theory, and you can read about each one in at most ten minutes. As we live in the era of Twitter, all the conjectures we state are 140 characters or fewer (so ‘‘minute’’ here has a double meaning). We might even call these sketchy tweets, as we present examples for each conjecture that you can doodle on as you read. Hamming also references ambiguity: good researchers can work both on proving and disproving the same statement, so we approach the conjectures with an open mind. He also mentions that a good approach is to reframe the problem, and change the point of view. One example from Vizing’s Conjecture (which is discussed as our second-to-last conjecture in the following text), is the three-page paper [2] which, with a new way of thinking, reduced most of the published work of twenty years to a corollary of its main result! Given the size of modern graph theory, with its many smaller subfields (such as structural graph theory, random graphs, topological graph theory, graph algorithms, spectral graph theory, graph minors, and graph homomorphisms, to name a few), it would be impossible to list all, or even the bulk of the conjectures in the field. We are content instead to focus on a few family jewels, which have an intrinsic beauty and have provided some challenges for graph-theorists for at least two decades. There is something for everyone here, from undergraduate students taking their first course in graph theory, to seasoned researchers in the field. For additional reading on problems and conjectures in graph theory and other fields, see the Open Problem Garden maintained by IRMACS at Simon Fraser University [24]. We consider only finite and undirected graphs, with no multiple edges or loops (unless otherwise stated). We assume the reader has some basic familiarity with graphs and their terminology, including notions such as cycles, paths, complete graphs, complete bipartite graphs, vertex degrees, and connected graphs. We use the notation Cn for the cycle with n vertices, Pn for the path with n vertices, and Kn for the complete graph with n vertices. The complete bipartite graph with m and n vertices of the respective colors is denoted by Km,n. All the background we need can be found in any text in graph theory, such as those of Diestel [9] and West [41], or online (see for example [42]).

Two remarks on the Burr–Erdős conjecture

In this paper, we develop efficient exact and approximate algorithms for computing a maximum independent set in random graphs. In a random graph G, each pair of vertices are joined by an edge with a probability p, where p is a constant between 0 and 1. We show that a maximum independent set in a random graph that contains n vertices can be computed in expected computation time . In addition, we show that, with high probability, the parameterized independent set problem is fixed parameter tractable in random graphs and the maximum independent set in a random graph in n vertices can be approximated within a ratio of in expected polynomial time.

We discuss a new algorithmic type of problem in random graphs studying the minimum number of queries one has to ask about adjacency between pairs of vertices of a random graph in order to find a subgraph which possesses some target property with high probability. In this paper we focus on finding long paths in when for some fixed constant . This random graph is known to have typically linearly long paths. To have edges with high probability in one clearly needs to query at least pairs of vertices. Can we find a path of length economically, i.e., by querying roughly that many pairs? We argue that this is not possible and one needs to query significantly more pairs. We prove that any randomised algorithm which finds a path of length with at least constant probability in with must query at least pairs of vertices. This is tight up to the factor. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 71–85, 2017

Quasirandom Rumor Spreading on Expanders

We study the evolution of the chromatic number of a random intersection graph and show that, in a certain range of parameters, these random graphs can be colored optimally with high probability using different greedy algorithms. Experiments on real network data confirm the positive theoretical predictions and suggest that heuristics for the clique and the chromatic number can work hand in hand proving mutual optimality.

For a fixed graph $H$ with $t$ vertices, an $H$-factor of a graph $G$ with $n$ vertices, where $t$ divides $n$, is a collection of vertex disjoint (not necessarily induced) copies of $H$ in $G$ covering all vertices of $G$. We prove that for a fixed tree $T$ on $t$ vertices and $\epsilon>0$, the random graph $G_{n,p}$, with $n$ a multiple of $t$, with high probability contains a family of edge-disjoint $T$-factors covering all but an $\epsilon$-fraction of its edges, as long as $\epsilon^4 n p \gg \log^2 n$. Assuming stronger divisibility conditions, the edge probability can be taken down to $p>\frac{C\log n}{n}$. A similar packing result is proved also for pseudo-random graphs, defined in terms of their degrees and co-degrees.

For graphs G and H, an H‐coloring of G is a map from the vertices of G to the vertices of H that preserves edge adjacency. We consider the following extremal enumerative question: for a given H, which connected n‐vertex graph with minimum degree δ maximizes the number of H‐colorings? We show that for nonregular H and sufficiently large n, the complete bipartite graph is the unique maximizer. As a corollary, for nonregular H and sufficiently large n the graph is the unique k‐connected graph that maximizes the number of H‐colorings among all k‐connected graphs. Finally, we show that this conclusion does not hold for all regular H by exhibiting a connected n‐vertex graph with minimum degree δ that has more ‐colorings (for sufficiently large q and n) than .