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A conjecture by Aharoni and Berger states that every family of n matchings of size n + 1 in a bipartite multigraph contains a rainbow matching of size n. In this paper we prove that matching sizes of 3 2 + o(1) n suffice to guarantee such a rainbow matching, which is asymptotically the same bound as the best-known one in the case where we only aim to find a… (More)

In 1966 Gallai asked whether all longest paths in a connected graph have nonempty intersection. This is not true in general and various counterexamples have been found. However, the answer to Gallai's question is positive for several well-known classes of graphs, as for instance connected outerplanar graphs, connected split graphs, and 2-trees. A graph is… (More)

We consider biased (1 : b) Avoider-Enforcer games in the monotone and strict versions. In particular, we show that Avoider can keep his graph being a forest for every but maybe the last round of the game if b 200n ln n. By this we obtain essentially optimal upper bounds on the threshold biases for the non-planarity game, the non-k-colorability game, and the… (More)

article(s) in LSE Research Online to facilitate their private study or for non-commercial research. You may not engage in further distribution of the material or use it for any profit-making activities or any commercial gain. You may freely distribute the URL Abstract For each real γ > 0 and integers ∆ ≥ 2 and k ≥ 1, we prove that there exist constants β >… (More)

For each real γ > 0 and integers Δ ≥ 2 and k ≥ 1, we prove that there exist constants β > 0 and C > 0 such that for all p ≥ C(log n/n) 1/Δ the random graph G(n, p) asymptotically almost surely contains-even after an adversary deletes an arbitrary (1/k − γ)-fraction of the edges at every vertex-a copy of every n-vertex graph with maximum degree at most Δ,… (More)

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