Small maximal matchings of random cubic graphs

We consider the expected size of a smallest maximal matching of cubic graphs. Firstly, we present a randomized greedy algorithm for finding a small maximal matching of cubic graphs. We analyze the average‐case performance of this heuristic on random n‐vertex cubic graphs using differential equations. In this way, we prove that the expected size of the maximal matching returned by the algorithm is asymptotically almost surely (a.a.s.) less than 0.34623n. We also give an existence proof which shows that the size of a smallest maximal matching of a random n‐vertex cubic graph is a.a.s. less than 0.3214n. It is known that the size of a smallest maximal matching of a random n‐vertex cubic graph is a.a.s. larger than 0.3158n. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 293–323, 2009     

MAX k‐CUT and approximating the chromatic number of random graphs

We consider the MAX k‐CUT problem on random graphs G_n,p. First, we bound the probable weight of a MAX k‐CUT using probabilistic counting arguments and by analyzing a simple greedy heuristic. Then, we give an algorithm that approximates MAX k‐CUT in expected polynomial time, with approximation ratio 1 + O((np)^‐1/2). Our main technical tool is a new bound on the probable value of Frieze and Jerrum's semidefinite programming (SDP)‐relaxation of MAX k‐CUT on random graphs. To obtain this bound, we show that the value of the SDP is tightly concentrated. As a further application of our bound on the probable value of the SDP, we obtain an algorithm for approximating the chromatic number of G_n,p, 1/n ≤ p ≤ 0.99, within a factor of O((np)^1/2) in polynomial expected time, thereby answering a question of Krivelevich and Vu. We give similar algorithms for random regular graphs. The techniques for studying the SDP apply to a variety of SDP relaxations of further NP‐hard problems on random structures and may therefore be of independent interest. For instance, to bound the SDP we estimate the eigenvalues of random graphs with given degree sequences. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Merging percolation on Zd and classical random graphs: Phase transition

We study a random graph model which is a superposition of bond percolation on Z^d with parameter p, and a classical random graph G(n,c/n). We show that this model, being a homogeneous random graph, has a natural relation to the so‐called “rank 1 case” of inhomogeneous random graphs. This allows us to use the newly developed theory of inhomogeneous random graphs to describe the phase diagram on the set of parameters c ≥ 0 and 0 ≤ p < p_c, where p_c = p_c(d) is the critical probability for the bond percolation on Z^d. The phase transition is of second order as in the classical random graph. We find the scaled size of the largest connected component in the supercritical regime. We also provide a sharp upper bound for the largest connected component in the subcritical regime. The latter is a new result for inhomogeneous random graphs with unbounded kernels. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

On the cover time of random walks on graphs

This article deals with random walks on arbitrary graphs. We consider the cover time of finite graphs. That is, we study the expected time needed for a random walk on a finite graph to visit every vertex at least once. We establish an upper bound ofO(n ²) for the expectation of the cover time for regular (or nearly regular) graphs. We prove a lower bound of (n logn) for the expected cover time for trees. We present examples showing all our bounds to be tight.Mike Saks was supported by NSF-DMS87-03541 and by AFOSR-0271. Jeff Kahn was supported by MCS-83-01867 and by AFOSR-0271.     

Random walks on highly symmetric graphs

We consider uniform random walks on finite graphs withn nodes. When the hitting times are symmetric, the expected covering time is at least 1/2n logn-O(n log logn) uniformly over all such graphs. We also obtain bounds for the covering times in terms of the eigenvalues of the transition matrix of the Markov chain. For distance-regular graphs, a general lower bound of (n-1) logn is obtained. For hypercubes and binomial coefficient graphs, the limit law of the covering time is obtained as well.     

Space complexity of random formulae in resolution

We study the space complexity of refuting unsatisfiable random k-CNFs in the Resolution proof system. We prove that for Δ ≥ 1 and any ϵ > 0, with high probability a random k-CNF over n variables and Δn clauses requires resolution clause space of Ω(n/Δ^1+ϵ). For constant Δ, this gives us linear, optimal, lower bounds on the clause space. One consequence of this lower bound is the first lower bound for size of treelike resolution refutations of random 3-CNFs with clause density Δ ≫ n. This bound is nearly tight. Specifically, we show that with high probability, a random 3-CNF with Δn clauses requires treelike refutation size of exp(Ω(n/Δ^1+ϵ)), for any ϵ > 0. Our space lower bound is the consequence of three main contributions: (1) We introduce a 2-player Matching Game on bipartite graphs G to prove that there are no perfect matchings in G. (2) We reduce lower bounds for the clause space of a formula F in Resolution to lower bounds for the complexity of the game played on the bipartite graph G(F) associated with F. (3) We prove that the complexity of the game is large whenever G is an expander graph. Finally, a simple probabilistic analysis shows that for a random formula F, with high probability G(F) is an expander. We also extend our result to the case of G-PHP, a generalization of the Pigeonhole principle based on bipartite graphs G. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 92–109, 2003     

Generating unlabeled connected cubic planar graphs uniformly at random

We present an expected polynomial time algorithm to generate an unlabeled connected cubic planar graph uniformly at random. We first consider rooted connected cubic planar graphs, i.e., we count connected cubic planar graphs up to isomorphisms that fix a certain directed edge. Based on decompositions along the connectivity structure, we derive recurrence formulas for the exact number of rooted cubic planar graphs. This leads to rooted 3‐connected cubic planar graphs, which have a unique embedding on the sphere. Special care has to be taken for rooted graphs that have a sense‐reversing automorphism. Therefore we introduce the concept of colored networks, which stand in bijective correspondence to rooted 3‐connected cubic planar graphs with given symmetries. Colored networks can again be decomposed along the connectivity structure. For rooted 3‐connected cubic planar graphs embedded in the plane, we switch to the dual and count rooted triangulations. Since all these numbers can be evaluated in polynomial time using dynamic programming, rooted connected cubic planar graphs can be generated uniformly at random in polynomial time by inverting the decomposition along the connectivity structure. To generate connected cubic planar graphs without a root uniformly at random, we apply rejection sampling and obtain an expected polynomial time algorithm. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Total domination in 2‐connected graphs and in graphs with no induced 6‐cycles

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ_t(G) of G. It is known [J Graph Theory 35 (2000), 21–45] that if G is a connected graph of order n > 10 with minimum degree at least 2, then γ_t(G) ≤ 4n/7 and the (infinite family of) graphs of large order that achieve equality in this bound are characterized. In this article, we improve this upper bound of 4n/7 for 2‐connected graphs, as well as for connected graphs with no induced 6‐cycle. We prove that if G is a 2‐connected graph of order n > 18, then γ_t(G) ≤ 6n/11. Our proof is an interplay between graph theory and transversals in hypergraphs. We also prove that if G is a connected graph of order n > 18 with minimum degree at least 2 and no induced 6‐cycle, then γ_t(G) ≤ 6n/11. Both bounds are shown to be sharp. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 55–79, 2009     

Dominating and total dominating partitions in cubic graphs

In this paper, we continue the study of domination and total domination in cubic graphs. It is known [Henning M.A., Southey J., A note on graphs with disjoint dominating and total dominating sets, Ars Combin., 2008, 89, 159–162] that every cubic graph has a dominating set and a total dominating set which are disjoint. In this paper we show that every connected cubic graph on nvertices has a total dominating set whose complement contains a dominating set such that the cardinality of the total dominating set is at most (n+2)/2, and this bound is essentially best possible.     

Balanced allocation on graphs: A random walk approach

We propose algorithms for allocating n sequential balls into n bins that are interconnected as a d‐regular n‐vertex graph G, where d ≥ 3 can be any integer. In general, the algorithms proceeds in n succeeding rounds. Let ? > 0 be an integer, which is given as an input to the algorithms. In each round, ball 1 ≤ t ≤ n picks a node of G uniformly at random and performs a nonbacktracking random walk of length ? from the chosen node and simultaneously collects the load information of a subset of the visited nodes. It then allocates itself to one of them with the minimum load (ties are broken uniformly at random). For graphs with sufficiently large girths, we obtain upper and lower bounds for the maximum number of balls at any bin after allocating all n balls in terms of ?, with high probability.     

The n‐ordered graphs: A new graph class

For a positive integer n, we introduce the new graph class of n‐ordered graphs, which generalize partial n‐trees. Several characterizations are given for the finite n‐ordered graphs, including one via a combinatorial game. We introduce new countably infinite graphs R⁽ⁿ⁾, which we name the infinite random n‐ordered graphs. The graphs R⁽ⁿ⁾ play a crucial role in the theory of n‐ordered graphs, and are inspired by recent research on the web graph and the infinite random graph. We characterize R⁽ⁿ⁾ as a limit of a random process, and via an adjacency property and a certain folding operation. We prove that the induced subgraphs of R⁽ⁿ⁾ are exactly the countable n‐ordered graphs. We show that all countable groups embed in the automorphism group of R⁽ⁿ⁾. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 204–218, 2009     

The analysis of a nested dissection algorithm

Summary Nested dissection is an algorithm invented by Alan George for preserving sparsity in Gaussian elimination on symmetric positive definite matrices. Nested dissection can be viewed as a recursive divide-and-conquer algorithm on an undirected graph; it usesseparators in the graph, which are small sets of vertices whose removal divides the graph approximately in half. George and Liu gave an implementation of nested dissection that used a heuristic to find separators. Lipton and Tarjan gave an algorithm to findn ^1/2-separators in planar graphs and two-dimensional finite element graphs, and Lipton, Rose, and Tarjan used these separators in a modified version of nested dissection, guaranteeing bounds ofO (n logn) on fill andO(n ^3/2) on operation count. We analyze the combination of the original George-Liu nested dissection algorithm and the Lipton-Tarjan planar separator algorithm. This combination is interesting because it is easier to implement than the Lipton-Rose-Tarjan version, especially in the framework of existïng sparse matrix software. Using some topological graph theory, we proveO(n logn) fill andO(n ^3/2) operation count bounds for planar graphs, twodimensional finite element graphs, graphs of bounded genus, and graphs of bounded degree withn ^1/2-separators. For planar and finite element graphs, the leading constant factor is smaller than that in the Lipton-Rose-Tarjan analysis. We also construct a class of graphs withn ^1/2-separators for which our algorithm does not achieve anO(n logn) bound on fill.The work of this author was supported in part by the Hertz Foundation under a graduate fellowship and by the National Science Foundation under Grant MCS 82-02948The work of this author was supported in part by the National Science Foundation under Grant MCS 78-26858 and by the Office of Naval Research under Contract N00014-76-C-0688     

For most large underdetermined systems of linear equations the minimal 𝓁1‐norm solution is also the sparsest solution

We consider linear equations y = Φx where y is a given vector in ?ⁿ and Φ is a given n × m matrix with n < m ≤ τn, and we wish to solve for x ∈ ?^m. We suppose that the columns of Φ are normalized to the unit ??₂‐norm, and we place uniform measure on such Φ. We prove the existence of ρ = ρ(τ) > 0 so that for large n and for all Φ's except a negligible fraction, the following property holds: For every y having a representation y = Φx₀ by a coefficient vector x₀ ∈ ?^m with fewer than ρ · n nonzeros, the solution x₁ of the ??₁‐minimization problem is unique and equal to x₀. In contrast, heuristic attempts to sparsely solve such systems—greedy algorithms and thresholding—perform poorly in this challenging setting. The techniques include the use of random proportional embeddings and almost‐spherical sections in Banach space theory, and deviation bounds for the eigenvalues of random Wishart matrices. © 2006 Wiley Periodicals, Inc.     

On the probability of independent sets in random graphs

Let k be the asymptotic value of the independence number of the random graph G(n, p). We prove that if the edge probability p(n) satisfies p(n) ? n^?2/5ln^6/5n then the probability that G(n, p) does not contain an independent set of size k ? c, for some absolute constant c > 0, is at most exp{?cn²/(k⁴p)}. We also show that the obtained exponent is tight up to logarithmic factors, and apply our result to obtain new bounds on the choice number of random graphs. We also discuss a general setting where our approach can be applied to provide an exponential bound on the probability of certain events in product probability spaces. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 22: 1–14, 2003     

On cubic non‐Cayley vertex‐transitive graphs

In 1983, the second author [D. Maru?i?, Ars Combinatoria 16B (1983), 297–302] asked for which positive integers n there exists a non‐Cayley vertex‐transitive graph on n vertices. (The term non‐Cayley numbers has later been given to such integers.) Motivated by this problem, Feng [Discrete Math 248 (2002), 265–269] asked to determine the smallest valency ?(n) among valencies of non‐Cayley vertex‐transitive graphs of order n. As cycles are clearly Cayley graphs, ?(n)?3 for any non‐Cayley number n. In this paper a goal is set to determine those non‐Cayley numbers n for which ?(n) = 3, and among the latter to determine those for which the generalized Petersen graphs are the only non‐Cayley vertex‐transitive graphs of order n. It is known that for a prime p every vertex‐transitive graph of order p, p² or p³ is a Cayley graph, and that, with the exception of the Coxeter graph, every cubic non‐Cayley vertex‐transitive graph of order 2p, 4p or 2p² is a generalized Petersen graph. In this paper the next natural step is taken by proving that every cubic non‐Cayley vertex‐transitive graph of order 4p², p>7 a prime, is a generalized Petersen graph. In addition, cubic non‐Cayley vertex‐transitive graphs of order 2p^k, where p>7 is a prime and k?p, are characterized. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 77–95, 2012     

Random graphs on surfaces

Counting labelled planar graphs, and typical properties of random labelled planar graphs, have received much attention recently. We start the process here of extending these investigations to graphs embeddable on any fixed surface S. In particular we show that the labelled graphs embeddable on S have the same growth constant as for planar graphs, and the same holds for unlabelled graphs. Also, if we pick a graph uniformly at random from the graphs embeddable on S which have vertex set {1,…,n}, then with probability tending to 1 as n→∞, this random graph either is connected or consists of one giant component together with a few nodes in small planar components.     

On two Hamilton cycle problems in random graphs

We study two problems related to the existence of Hamilton cycles in random graphs. The first question relates to the number of edge disjoint Hamilton cycles that the random graph G _n,p contains. δ(G)/2 is an upper bound and we show that if p ≤ (1 + o(1)) ln n/n then this upper bound is tight whp. The second question relates to how many edges can be adversarially removed from G _n,p without destroying Hamiltonicity. We show that if p ≥ K ln n/n then there exists a constant α > 0 such that whp G − H is Hamiltonian for all choices of H as an n-vertex graph with maximum degree Δ(H) ≤ αK ln n. Research supported in part by NSF grant CCR-0200945. Research supported in part by USA-Israel BSF Grant 2002-133 and by grant 526/05 from the Israel Science Foundation.     

Estimates of coefficients of chromatic polynomials and numbers of cliques of (c,n,m)‐graphs

This paper is mainly concerned with classes of simple graphs with exactly c connected components, n vertices and m edges, for fixed c,n,m ∈ ?. We find an optimal lower bound for the ith coefficient of the chromatic polynomial of a graph in such a class and also an optimal upper bound for the number of j‐cliques contained in such a graph. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 81–94, 2003     

Largest planar matching in random bipartite graphs

For a distribution ?? over labeled bipartite (multi) graphs G = (W, M, E), |W| = |M| = n, let L(n) denote the size of the largest planar matching of G (here W and M are posets drawn on the plane as two ordered rows of nodes and edges are drawn as straight lines). We study the asymptotic (in n) behavior of L(n) for different distributions ??. Two interesting instances of this problem are Ulam's longest increasing subsequence problem and the longest common subsequence problem. We focus on the case where ?? is the uniform distribution over the k‐regular bipartite graphs on W and M. For k = o(n^1/4), we establish that $L(n) \slash \sqrt{kn}$ tends to 2 in probability when n → ∞. Convergence in mean is also studied. Furthermore, we show that if each of the n² possible edges between W and M are chosen independently with probability 0 < p < 1, then L(n)/n tends to a constant γ_p in probability and in mean when n → ∞. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 162–181, 2002     

Planar graphs: Random walks and bipartiteness testing

We initiates the study of property testing in arbitrary planar graphs. We prove that bipartiteness can be tested in constant time, improving on the previous bound of for graphs on n vertices. The constant‐time testability was only known for planar graphs with bounded degree. Our algorithm is based on random walks. Since planar graphs have good separators, that is, bad expansion, our analysis diverges from standard techniques that involve the fast convergence of random walks on expanders. We reduce the problem to the task of detecting an odd‐parity cycle in a multigraph induced by constant‐length cycles. We iteratively reduce the length of cycles while preserving the detection probability, until the multigraph collapses to a collection of easily discoverable self‐loops. Our approach extends to arbitrary minor‐free graphs. We also believe that our techniques will find applications to testing other properties in arbitrary minor‐free graphs.