Component behaviour and excess of random bipartite graphs near the critical point

ABSTRACT. The binomial random bipartite graph G(n,n, p) is the random graph formed by taking two partition classes of size n and including each edge between them independently with probability p. It is known that this model exhibits a similar phase transition as that of the binomial random graph G(n, p) as p passes the critical point of 1 n . We study the component structure of this model near to the critical point. We show that, as with G(n, p), for an appropriate range of p there is a unique ‘giant’ component and we determine asymptotically its order and excess. We also give more precise results for the distribution of the number of components of a fixed order in this range of p. These results rely on new bounds for the number of bipartite graphs with a fixed number of vertices and edges, which we also derive.

[1]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[2]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[3]  Tomasz Luczak,et al.  Cycles in a Random Graph Near the Critical Point , 1991, Random Struct. Algorithms.

[4]  Tony Johansson.  The giant component of the random bipartite graph , 2012 .

[5]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[6]  Mihyun Kang,et al.  Planarity and Genus of Sparse Random Bipartite Graphs , 2020, SIAM J. Discret. Math..

[7]  H. I. Scoins,et al.  The number of trees with nodes of alternate parity , 1962, Mathematical Proceedings of the Cambridge Philosophical Society.

[8]  Tomasz Luczak.  On the Number of Sparse Connected Graphs , 1990, Random Struct. Algorithms.

[9]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[10]  Michael Krivelevich,et al.  The Genus of the Erdös-Rényi Random Graph and the Fragile Genus Property , 2018, AofA.

[11]  Mihyun Kang,et al.  The Phase Transition in Multitype Binomial Random Graphs , 2015, SIAM J. Discret. Math..

[12]  Brendan D. McKay,et al.  The Asymptotic Number of Labeled Connected Graphs with a Given Number of Vertices and Edges , 1990, Random Struct. Algorithms.

[13]  T. E. Harris.  A lower bound for the critical probability in a certain percolation process , 1960, Mathematical Proceedings of the Cambridge Philosophical Society.

[14]  Tomasz Luczak,et al.  Component Behavior Near the Critical Point of the Random Graph Process , 1990, Random Struct. Algorithms.

[15]  Bojan Mohar,et al.  The Genus of a Random Bipartite Graph , 2019, Canadian Journal of Mathematics.

[16]  Yoshiharu Kohayakawa,et al.  The Evaluation of Random Subgraphs of the Cube , 1992, Random Struct. Algorithms.