In a recent paper Velleman and Warrington analyzed the expected values of some of the parameters in a memory game, namely, the length of the game, the waiting time for the first match, and the number of lucky moves. In this paper we continue this direction of investigation and obtain the limiting distributions of those parameters. More specifically, we prove that when suitably normalized, these quantities converge in distribution to a normal, Rayleigh, and Poisson random variable, respectively.We also make a connection between the memory game and one of the models of preferential attachment graphs. In particular, as a by-product of our methods we obtain simpler proofs (although without rate of convergence) of some of the results of Peköz, Röllin, and Ross on the joint limiting distributions of the degrees of the first few vertices in preferential attachment graphs.For proving that the length of the game is asymptotically normal, our main technical tool is a limit result for the joint distribution of the number of balls in a multi-type generalized Pólya urn model.

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We consider the problem of designing succinct data structures for interval graphs with n vertices while supporting degree, adjacency, neighborhood and shortest path queries in optimal time in the Θ(log n)bit 1 word RAM model. The degree query reports the number of incident edges to a given vertex in constant time, the adjacency query returns true if there is an edge between two vertices in constant time, the neighborhood query reports the set of all adjacent vertices in time proportional to the degree of the queried vertex, and the shortest path query returns a shortest path in time proportional to its length, thus the running times of these queries are optimal. Towards showing succinctness, we first show that at least n log n − 2n log log n − O(n) bits are necessary to represent any unlabeled interval graph G with n vertices, answering an open problem of Yang and Pippenger [Proc. Amer. Math. Soc. 2017]. This is augmented by a data structure of size n log n + O(n) bits while supporting not only the aforementioned queries optimally but also capable of executing various combinatorial algorithms (like proper coloring, maximum independent set etc.) on the input interval graph efficiently. Finally, we extend our ideas to other variants of interval graphs, for example, proper/unit interval graphs, k-proper and k-improper interval graphs, and circular-arc graphs, and design succinct/compact data structures for these graph classes as well along with supporting queries on them efficiently.

A permutation σ of [n] induces a graph Gσ on [n] -its edges are inversion pairs in σ. The graph Gσ is connected if and only if σ is indecomposable. Let σ(n, m) denote a permutation chosen uniformly at random among all permutations of [n] with m inversions. Let p(n, m) be the common value for the probabilities P(σ(n, m) is indecomposable) and P(G σ(n,m) is connected). We prove that p(n, m) is non-decreasing with m by constructing a Markov process {σ(n, m)} in which σ(n, m + 1) is obtained by increasing one of the components of the inversion sequence of σ(n, m) by one. We show that, with probability approaching 1, G σ(n,m) becomes connected for m asymptotic to mn = (6/π 2 )n log n. We also find the asymptotic sizes of the largest and smallest components when the number of edges is moderately below the threshold mn.1991 Mathematics Subject Classification. 05A05; 60C05.

A permutation w gives rise to a graph Gw; the vertices of Gw are the letters in the permutation and the edges of Gw are the inversions of w. We find that the number of trees among permutation graphs with n vertices is 2 n−2 for n ≥ 2. We then study Tn, a uniformly random tree from this set of trees. In particular, we study the number of vertices of a given degree in Tn, the maximum degree in Tn, the diameter of Tn, and the domination number of Tn. Denoting the number of degree-k vertices in Tn by D k , we find that (D1, . . . , Dm) converges to a normal distribution for any fixed m as n → ∞. The vertex domination number of Tn is also asymptotically normally distributed as n → ∞. The diameter of Tn shifted by −2 is binomially distributed with parameters n − 3 and 1/2. Finally, we find the asymptotic distribution of the maximum degree in Tn, which is concentrated around log 2 n.