Directed paths: from Ramsey to Ruzsa and Szemeredi

<p>Starting from an innocent Ramsey-theoretic question regarding directed paths in tournaments, we discover a series of rich and surprising connections that lead into the theory around a fundamental problem in Combinatorics: the Ruzsa-Szemeredi induced matching problem. Using these relationships, we prove that every coloring of the edges of the transitive n-vertex tournament using three colors contains a directed path of length at least n−−√⋅e^log∗n which entirely avoids some color. We also expose connections to a family of constructions for Ramsey tournaments, and introduce and resolve some natural generalizations of the Ruzsa-Szemeredi problem which we encounter through our investigation.</p

Diameter critical graphs

<p>A graph is called diameter-k-critical if its diameter is k, and the removal of any edge strictly increases the diameter. In this paper, we prove several results related to a conjecture often attributed to Murty and Simon, regarding the maximum number of edges that any diameter-k-critical graph can have. In particular, we disprove a longstanding conjecture of Caccetta and H¨aggkvist (that in every diameter-2-critical graph, the average edge-degree is at most the number of vertices), which promised to completely solve the extremal problem for diameter-2-critical graphs.</p> <p>On the other hand, we prove that the same claim holds for all higher diameters, and is asymptotically tight, resolving the average edge-degree question in all cases except diameter-2. We also apply our techniques to prove several bounds for the original extremal question, including the correct asymptotic bound for diameter-k-critical graphs, and an upper bound of ( 1/6 +o(1))n² for the number of edges in a diameter-3-critical graph.</p

On a problem of Erdős and Rothschild on edges in triangles

Erdos and Rothschild asked to estimate the maximum number, denoted by h(n, c), such that every n-vertex graph with at least cn2 edges, each of which is contained in at least one triangle, must contain an edge that is in at least h(n, c) triangles. In particular, Erdos asked in 1987 to determine whether for every c > 0 there is ϵ > 0 such that h(n,c) > nϵ for all sufficiently large n. We prove that h(n, c) = n O(1/ log log n) for every fixed c 2/4 edges contains an edge that is in at least n/6 triangles.</p

Rainbow Hamilton cycles in random graphs

<p>One of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erdős-Rényi random graph <em>G</em>_<em>n,p</em> is around . Much research has been done to extend this to increasingly challenging random structures. In particular, a recent result by Frieze determined the asymptotic threshold for a loose Hamilton cycle in the random 3-uniform hypergraph by connecting 3-uniform hypergraphs to edge-colored graphs.</p> <p>In this work, we consider that setting of edge-colored graphs, and prove a result which achieves the best possible first order constant. Specifically, when the edges of <em>G</em>_<em>n,p</em> are randomly colored from a set of (1 + <em>o</em>(1))<em>n</em> colors, with , we show that one can almost always find a Hamilton cycle which has the additional property that all edges are distinctly colored (rainbow).</p

Hamiltonian increasing paths in random edge orderings

<p>Let f be an edge ordering of Kn: a bijection E(Kn) → {1, 2, . . . , n2}. For an edge e ∈ E(Kn), we call f(e) the label of e. An increasing path in Kn is a simple path (visiting each vertex at most once) such that the label on each edge is greater than the label on the previous edge. We let S(f) be the number of edges in the longest increasing path. Chvatal and Komlos raised the question of estimating m(n): the minimum value of S(f) over all orderings f of Kn. The best known bounds on m(n) are , due respectively to Graham and Kleitman, and to Calderbank, Chung, and Sturtevant. Although the problem is natural, it has seen essentially no progress for three decades.</p> <p>In this paper, we consider the average case, when the ordering is chosen uniformly at random. We discover the surprising result that in the random setting, S(f) often takes its maximum possible value of n − 1 (visiting all of the vertices with a Hamiltonian increasing path). We prove that this occurs with probability at least about 1/e. We also prove that with probability 1 − o(1), there is an increasing path of length at least 0.85n, suggesting that this Hamiltonian (or near-Hamiltonian) phenomenon may hold asymptotically almost surely.</p

Stochastic coalescence in logarithmic time

<p>The following distributed coalescence protocol was introduced by Dahlia Malkhi in 2006 motivated by applications in social networking. Initially there are n agents wishing to coalesce into one cluster via a decentralized stochastic process, where each round is as follows: every cluster flips a fair coin to dictate whether it is to issue or accept requests in this round. Issuing a request amounts to contacting a cluster randomly chosen <em>proportionally to its size</em>. A cluster accepting requests is to select an incoming one <em>uniformly</em> (if there are such) and merge with that cluster. Empirical results by Fernandess and Malkhi suggested the protocol concludes in O(logn) rounds with high probability, whereas numerical estimates by Oded Schramm, based on an ingenious analytic approximation, suggested that the coalescence time should be super-logarithmic.</p> <p>Our contribution is a rigorous study of the stochastic coalescence process with two consequences. First, we confirm that the above process indeed requires super-logarithmic time w.h.p., where the inefficient rounds are due to oversized clusters that occasionally develop. Second, we remedy this by showing that a simple modification produces an essentially optimal distributed protocol; if clusters favor their <em>smallest</em> incoming merge request then the process <em>does</em> terminate in O(logn) rounds w.h.p., and simulations show that the new protocol readily outperforms the original one. Our upper bound hinges on a potential function involving the logarithm of the number of clusters and the cluster-susceptibility, carefully chosen to form a supermartingale. The analysis of the lower bound builds upon the novel approach of Schramm which may find additional applications: rather than seeking a single parameter that controls the system behavior, instead one approximates the system by the Laplace transform of the entire cluster-size distribution.</p

Cops and robbers on planar directed graphs

<p>Aigner and Fromme initiated the systematic study of the cop number of a graph by proving the elegant and sharp result that in every connected planar graph, three cops are sufficient to win a natural pursuit game against a single robber. This game, introduced by Nowakowski and Winkler, is commonly known as Cops and Robbers in the combinatorial literature. We extend this study to directed planar graphs, and establish separation from the undirected setting. We exhibit a geometric construction which shows that a more sophisticated robber strategy can indefinitely evade three cops on a particular strongly connected planar directed graph</p

Computing with Voting Trees

<p>The classical paradox of social choice theory asserts that there is no fair way to deterministically select a winner in an election among more than two candidates; the only definite collective preferences are between individual pairs of candidates. Combinatorially, one may summarize this information with a graph-theoretic tournament on n vertices (one per candidate), placing an edge from u to v if u would beat v in an election between only those two candidates (no ties are permitted). One well-studied procedure for selecting a winner is to specify a complete binary tree whose leaves are labeled by the candidates, and evaluate it by running pairwise elections between the pairs of leaves, sending the winners to successive rounds of pairwise elections which ultimately terminate with a single winner. This structure is called a voting tree.</p> <p>Much research has investigated which functions on tournaments are computable in this way. Fischer, Procaccia, and Samorodnitsky quantitatively studied the computability of the Copeland rule, which returns a vertex of maximum out-degree in the given tournament. Perhaps surprisingly, the best previously known voting tree could only guarantee a returned out-degree of at least log2 n, despite the fact that every tournament has a vertex of degree at least n−1 2 . In this paper, we present three constructions, the first of which substantially improves this guarantee to Θ(√ n). The other two demonstrate the richness of the voting tree universe, with a tree that resists manipulation, and a tree which implements arithmetic modulo three.</p

Bisections of graphs

<p>A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differ by at most 1, and its size is the number of edges which go across the two parts. In this paper, motivated by several questions and conjectures of Bollobás and Scott, we study maximum bisections of graphs. First, we extend the classical Edwards bound on maximum cuts to bisections. A simple corollary of our result implies that every graph on <em>n</em> vertices and <em>m </em> edges with no isolated vertices, and maximum degree at mostn/3+1, admits a bisection of size at least m/2+n/6. Then using the tools that we developed to extend Edwardsʼs bound, we prove a judicious bisection result which states that graphs with large minimum degree have a bisection in which both parts span relatively few edges. A special case of this general theorem answers a conjecture of Bollobás and Scott, and shows that every graph on <em>n</em> vertices and <em>m </em> edges of minimum degree at least 2 admits a bisection in which the number of edges in each part is at most(1/3+o(1))m. We also present several other results on bisections of graphs.</p

Judicious partitions of directed graphs

<p>The area of judicious partitioning considers the general family of partitioning problems in which one seeks to optimize several parameters simultaneously, and these problems have been widely studied in various combinatorial contexts. In this paper, we study essentially the most fundamental judicious partitioning problem for directed graphs, which naturally extends the classical Max Cut problem to this setting: we seek bipartitions in which many edges cross in each direction. It is easy to see that a minimum outdegree condition is required in order for the problem to be nontrivial, and we prove that every directed graph with m edges and minimum outdegree at least two admits a bipartition in which at least ( 1/6 + o(1))m edges cross in each direction. We also prove that if the minimum outdegree is at least three, then the constant can be increased to 1/5 . If the minimum outdegree tends to infinity with n, then the constant increases to 1/4 . All of these constants are best-possible, and provide asymptotic answers to a question of Alex Scott</p