Generating random graphs in biased Maker-Breaker games

Abstract

We present a general approach connecting biased Maker-Breaker games and problems about local resilience in random graphs. We utilize this approach to prove new results and also to derive some known results about biased Maker-Breaker games. In particular, we show that for b = o ( √ n), Maker can build a pancyclic graph (that is, a graph that contains cycles of every possible length) while playing a (1 : b) game on E(K n ). As another application, we show that for b = Θ (n/ ln n), playing a (1 : b) game on E(K n ), Maker can build a graph which contains copies of all spanning trees having maximum degree ∆ = O(1) with a bare path of linear length (a bare path in a tree T is a path with all interior vertices of degree exactly two in T ).

Rainbow Hamilton cycles in random graphs and hypergraphs

“…, consider a path P of G 1 ∪ {e} of maximal length which contains e as an edge. A non-edge e ′ of G 1 is called an e-booster if G 1 ∪ {e, e ′ } has fewer connected components than G 1 ∪ {e} has, or contains a path P ′ which passes through e and which is longer than P , or that G 1 ∪ {e, e ′ } contains a Hamilton cycle that uses e. The following lemma is from [9] and shows that every connected and non-Hamiltonian graph G 1 with "good" expansion properties has many e-boosters for every possible e. Lemma 2.20. Let G 1 be a connected graph for which |N G 1 (X) \ X| ≥ 2|X| + 2 holds for every subset X ⊆ V (G 1 ) of size |X| ≤ k. Then, for every pair e ∈ V (G 1 ) 2 such that G 1 ∪ {e} does not contain a Hamilton cycle which uses the edge e, the number of e-boosters for G 1 is at least (k + 1) 2 /2.…”

Robustness of graph properties

“…• In [37] the authors made a very interesting connection between the local resilience in random graphs and winning strategies in biased Maker-Breaker games.…”

Finding Hamilton cycles in random graphs with few queries

“…A non‐edge f of H is called an e ‐booster if contains a path Q which passes through e and which is longer than P or if contains a Hamilton cycle that uses e . The following lemma appears in and shows that every connected and non‐Hamiltonian graph G satisfying certain expansion properties has many e ‐boosters for every possible e . Lemma Let H be a connected graph for which holds for every subset of size .…”

“…Given a graph H and a pair e ∈ V (H) 2 , consider a path P of H ∪ {e} of maximal length which contains e as an edge. A non-edge f of H is called an e-booster if H ∪{e, f } contains a path Q which passes through e and which is longer than P or if H ∪{e, f } contains a Hamilton cycle that uses e. The following lemma appears in [8] and shows that every connected and non-Hamiltonian graph G satisfying certain expansion properties has many e-boosters for every possible e.…”