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We study the arboricity A and the maximum number T of edge‐disjoint spanning trees of the classical random graph G(n,p). For all p(n)∈[0,1], we show that, with high probability, T is precisely the minimum of normalδ and true⌊m/(n−1)true⌋, where normalδ is the minimum degree of the graph and m denotes the number of edges. Moreover, we explicitly determine a sharp threshold value for p such that the following holds. Above this threshold, T equals true⌊m/(n−1)true⌋ and A equals true⌈m/(n−1)true⌉. Below this threshold, T equals normalδ, and we give a two‐value concentration result for the arboricity A in that range. Finally, we include a stronger version of these results in the context of the random graph process where the edges are randomly added one by one. A direct application of our result gives a sharp threshold for the maximum load being at most k in the two‐choice load balancing problem, where k→∞.

We study the density of fixed strongly connected subtournaments on 5 vertices in large tournaments. We determine the maximum density asymptotically for five tournaments as well as unique extremal sequences for each tournament. As a byproduct we also characterize tournaments that are recursive blow-ups of a 3-cycle as tournaments that avoid three specific tournaments of size 5.

We study the arboricity A and the maximum number T of edge-disjoint spanning trees of the classical random graph G (n, p). For all p(n) ∈ [0, 1], we show that, with high probability, T is precisely the minimum between δ and m/(n − 1) , where δ is the smallest degree of the graph and m denotes the number of edges. Moreover, we explicitly determine a sharp threshold value for p such that: above this threshold, T equals m/(n − 1) and A equals m/(n − 1) ; and below this threshold, T equals δ, and we give a two-value concentration result for the arboricity A in that range. Finally, we include a stronger version of these results in the context of the random graph process where the edges are sequentially added one by one. A direct application of our result gives a sharp threshold for the maximum load being at most k in the two-choice load balancing problem, where k → ∞. IntroductionSTP number and arboricity: The spanning-tree packing (STP) number of a graph is the maximum number of edge-disjoint spanning trees it contains. Computing this parameter is a very classical problem in combinatorial optimization. One of the earliest results on the STP number is a min-max relation proved by Tutte [33] and Nash-Williams [27]: the STP number of a graph is the minimum value, ranging over all partitions P of the vertex set, of the ratio (rounded down) between the number of edges across P (i.e. edges with ends lying in different classes of P) and |P| − 1. This characterisation has important consequences in computer science, where the STP number has been used as a measure of network vulnerability in case of attack or edge failure (see [21,9]). Intuitively speaking, it provides information about the number of

We consider the following iterative construction of a random planar triangulation.Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. After n -3 steps, we obtain a random triangulated plane graph with n vertices, which is called a Random Apollonian Network (RAN). We show that asymptotically almost surely (a.a.s.) a longest path in a RAN has length o(n), refuting a conjecture of Frieze and Tsourakakis. We also show that a RAN always has a cycle (and thus a path) of length (2n − 5) log 2/ log 3 , and that the expected length of its longest cycles (and paths) is n 0.88 . Finally, we prove that a.a.s. the diameter of a RAN is asymptotic to c log n, where c ≈ 1.668 is the solution of an explicit equation.and we say a.a.s. X = o(f (n)) if for every fixed ε > 0, lim n→∞ P X ≤ εf (n) = 1.The authors of [9] conjecture in their concluding remarks that a.a.s. a RAN has a path of length (n). We refute this conjecture by showing the following theorem. Let L m be a random variable denoting the number of vertices in a longest path in a RAN with m faces. Theorem 1.1. A.a.s. we have L m = o(m).Recall that a RAN on n vertices has 2n − 5 faces, so Theorem 1.1 implies that a.a.s. a RAN does not have a path of length (n).We also prove lower bounds for L m deterministically, and its expected value in a RAN. In fact, we will prove lower bounds for C m and E [C m ], where C m denotes the number of Random Structures and Algorithms