The Biased Odd Cycle Game

The Biased Odd Cycle Game

In this paper we consider biased Maker-Breaker games played on the edge set of a given graph G. We prove that for every δ > 0 and large enough n, there exists a constant k for which if δ(G) ≥ δn and χ(G) ≥ k, then Maker can build an odd cycle in the (1 : b) game for b = O ( n log n ) . We also consider the analogous game where Maker and Breaker claim vertices instead of edges. This is a special...

Parallel Repetition of the Odd Cycle Game

Higher powers of the Odd Cycle Game has been the focus of recent investigations [3,4]. In [4] it was shown that if we repeat the game d times in parallel, the winning probability is upper bounded by 1 −Ω( √

Cobounding Odd Cycle Colorings

We give a very short self-contained combinatorial proof of the Babson-Kozlov conjecture, by presenting a cochain whose coboundary is the desired power of the characteristic class.

Doubly Biased Maker-Breaker Connectivity Game

In this paper we study the (a : b) Maker-Breaker Connectivity game, played on the edge set of the complete graph on n vertices. We determine the winner for almost all values of a and b.

Nesting of cycle systems of odd length