We prove that one can perfectly pack degenerate graphs into complete or dense n-vertex quasirandom graphs, provided that all the degenerate graphs have maximum degree o n log n , and in addition Ω(n) of them have at most 1 − Ω(1) n vertices and Ω(n) leaves. This proves Ringel's conjecture and the Gyárfás Tree Packing Conjecture for all but an exponentially small fraction of trees (or sequences of trees, respectively).

Given graphs G and H and a positive integer q, say that G is q‐Ramsey for H, denoted G→(H)q, if every q‐coloring of the edges of G contains a monochromatic copy of H. The size‐Ramsey number truerˆ(H) of a graph H is defined to be truerˆ(H)=min{∣E(G)∣:G→(H)2}. Answering a question of Conlon, we prove that, for every fixed k, we have truerˆ(Pnk)=O(n), where Pnk is the kth power of the n‐vertex path Pn (ie, the graph with vertex set V(Pn) and all edges {u,v} such that the distance between u and v in Pn is at most k). Our proof is probabilistic, but can also be made constructive.

Abstract. For a tree T on n vertices, we study the Maker-Breaker game, played on the edge set of the complete graph on n vertices, which Maker wins as soon as the graph she builds contains a copy of T . We prove that if T has bounded maximum degree and n is sufficiently large, then Maker can win this game within n + 1 moves. Moreover, we prove that Maker can build almost every tree on n vertices in n − 1 moves and provide nontrivial examples of families of trees which Maker cannot build in n − 1 moves.

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 case we only aim to find a rainbow matching of size n − 1. This improves previous results

We consider biased (1 : b) Walker-Breaker games: Walker and Breaker alternately claim edges of the complete graph K n , Walker taking one edge and Breaker claiming b edges in each round, with the constraint that Walker needs to choose her edges according to a walk. As questioned in a paper by Espig, Frieze, Krivelevich and Pegden, we study how long a cycle Walker is able to create and for which biases b Walker has a chance to create a cycle of given constant length. n 2 + 1 rounds, as shown by Hefetz, Krivelevich, Stojaković and Szabó [14]. She can create a Hamilton cycle within n + 1 rounds [16], and for k ≥ 2 she can create a k-connected spanning subgraph within ⌊ kn 2 ⌋ + 1 rounds [10]. So, for any of these structures the number of rounds she needs to play is at most one larger than the minimal size of a winning set.Due to this overwhelming power of Maker in these kinds of games, it is natural to consider variations that help to increase Breaker's power. One natural way is to increase Breaker's