Asymptotic random graph intuition for the biased connectivity game

Abstract

ABSTRACT:We study biased Maker/Breaker games on the edges of the complete graph, as introduced by Chvátal and Erdős. We show that Maker, occupying one edge in each of his turns, can build a spanning tree, even if Breaker occupies bedges in each turn. This improves a result of Beck, and is asymptotically best possible as witnessed by the Breaker-strategy of Chvátal and Erdős. We also give a strategy for Maker to occupy a graph with minimum degree c (where c = c(n) is a slowly growing function of n) while playing against a Breaker who takes b ≤ (1 − o(1)) · n ln n edges in each turn. This result improves earlier bounds by Krivelevich and Szabó. Both of our results support the surprising random graph intuition: the threshold bias is asymptotically the same for the game played by two "clever" players and the game played by two "random" players.

The critical bias for the Hamiltonicity game is (1+𝑜(1))𝑛/ln𝑛

“…Now we can describe the main tool of our proof, a recent result of Gebauer and Szabó, who analyzed in [6] the biased minimum degree game. For our goals, it will suffice to specialize their analysis to the game where Maker's goal is to reach a graph of minimum degree at least 12.…”

“…The proof of Lemma 3 is a straightforward modification of the proof of Theorem 1.2 of [6]. More specifically, the argument of [6] can be used to analyze a slightly different game in which Breaker wins if he accumulates at least (1 − δ)n edges at a vertex whose Maker degree is still less than 12.…”

“…In a relevant development, Gebauer and Szabó showed recently in [6] that the critical bias for the connectivity game on K n (where Maker wins if and only if he creates a spanning tree from his edges by the end of the game) is asymptotically equal to n/ ln n. We will rely extensively on some of their results and approaches here.…”

A threshold for the Maker-Breaker clique game

“…For the connectivity game, it was shown by Chvátal and Erdős [9] and Gebauer and Szabó [11] that the threshold bias is b T = (1 + o(1)) n log n . The result of Krivelevich [16] gives the leading term of the threshold bias for the Hamilton cycle game, b H = (1 + o(1)) n log n .…”

Fast embedding of spanning trees in biased Maker-Breaker games

“…Indeed, it was proved in [11] that, playing a (1 : 1) game on E(K n ), Maker can claim all edges of a Hamilton path of K n in n − 1 moves. Moreover, if Maker just wants to build a connected spanning graph, that is, he does not have to declare in advance which spanning tree he intends to build, then he can do so in n − 1 moves even in a (1 : (1 − ε)n/ log n) game (see [8]). On the other hand, it was conjectured by Beck [3] and subsequently proved by Bednarska [5] that, playing a (1 : q) game on E(K n ), where q ≥ cn for an arbitrarily small constant c > 0, Maker cannot build a complete binary tree on q vertices in optimal time, that is, in q − 1 moves.…”