Arboricity and spanning‐tree packing in random graphs
Pu Gao, Xavier Pérez‐Giménez, Cristiane M. Sato · Random Structures and Algorithms · 2017
Abstract We study the arboricity and the maximum number of edge‐disjoint spanning trees of the classical random graph . For all , we show that, with high probability, is precisely the minimum of and , where is the minimum degree of the graph and denotes the number of edges. Moreover, we explicitly determine a sharp threshold value for such that the following holds. Above this threshold, equals and equals . Below this threshold, equals , and we give a two‐value concentration result for the arboricity 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 in the two‐choice load balancing problem, where .