Positional games on randomly perturbed graphs

Abstract

Maker-Breaker games are played on a hypergraph pX, F q, where F Ď 2 X denotes the family of winning sets. Both players alternately claim a predefined amount of edges (called bias) from the board X, and Maker wins the game if she is able to occupy any winning set F P F . These games are well studied when played on the complete graph K n or on a random graph G n,p .In this paper we consider Maker-Breaker games played on randomly perturbed graphs instead.These graphs consist of the union of a deterministic graph G α with minimum degree at least αn and a binomial random graph G n,p . Depending on α and Breaker's bias b we determine the order of the threshold probability for winning the Hamiltonicity game and the k-connectivity game on G α Y G n,p , and we discuss the H-game when b " 1. Furthermore, we give optimal results for the Waiter-Client versions of all mentioned games. OP was supported by the DFG (Grant PA 3513/1-1).1 2 DENNIS CLEMENS, FABIAN HAMANN, YANNICK MOGGE, AND OLAF PARCZYK minimum degree at least n`k´1 2 is k-vertex-connected. Again this bound is seen to be sharp by looking at the following example: take the vertex disjoint union of three cliques of sizes k ´1, t n´k`1 2 u and r n´k`1 2 s, respectively, and add all edges between the first clique and both of the other cliques. For the containment of a fixed graph H a sufficient minimum degree condition can be derived from Turán's Theorem [46] with regularity, or from the following theorem of Erdős and Stone [20]: for any fixed graph H with chromatic number r ą 2, any n-vertex graph with minimum degree `r´2 r´1 `op1q ˘n contains a copy of H provided that n is large. This bound is seen to be sharp by considering a Turán graph on n vertices with r ´1 vertex classes, i.e. an pr ´1q-partite graph with all class sizes differing at most by 1. For bipartite graphs H any linear minimum degree is sufficient.1.2. Random graphs. Let G n,p be the binomial random graph model, i.e. the model of n-vertex random graphs, where each edge is present with probability p independently of all others. For simplicity, we will often write G n,p also for a graph G " G n,p drawn according to this distribution.Given some graph property P, we are interested in a probability function p " ppPq such that the following holds: when p " ωppq, then G n,p satisfies P asymptotically almost surely (a.a.s.), i.e. with probability tending to one as n tends to infinity. When additionally it holds that G n,p a.a.s. does not have property P when p " oppq, we call p a threshold probability for the property P. The existence of such a threshold is known for all monotone graph properties P [12]. If the same conclusion even holds with p ě p1 `εqp and p ď p1 ´εqp for any constant ε ą 0, then we call p a sharp threshold probability for the property P.Posá [43] and Korshunov [36] independently proved that G n,p has a sharp threshold for containing