Arboricity and spanning-tree packing in random graphs with an application to load balancing

Abstract

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

Optimal path and cycle decompositions of dense quasirandom graphs

“…A related result of Gao, Pérez-Giménez and Sato [11] determines the arboricity and spanning tree packing number of G n,p . Optimal results on packing Hamilton cycles in G n,p which together cover essentially the whole range of p were proven in [17,19].…”

Arboricity and spanning‐tree packing in random graphs

“…Consider the constantˆ = 0.1/(γ − 0.8) ∈ (0, 1], and defineα = (1 +ˆ )α. Recall that both α andα are functions of x = p(n − 1)/ log n. Then, using (22) and the fact that ≤ 0.1/(x − 0.8), we obtain…”

“…Let x = p(n − 1)/ log n and define α = a(x) ∈ (0, 1 − η). From (22) and by Lemma 19 (ii), we can bound…”

“… An extended abstract of this article has appeared in the Proceedings of the 25th Annual ACM‐SIAM Symposium on Discrete Algorithms (Soda'14). See . …”

Sandwiching a densest subgraph by consecutive cores

“…G k contains no subgraph with density greater than k. The following proposition follows by letting → 0 and by Theorem 2.1 together with a simple coupling argument (to deal with the case that the density of the k-core is between k − 1 − and k − 1 + ). For the complete proof of the proposition, refer to [7,Corollary 31].…”