Decomposing almost complete graphs by random trees

Preprin

Decomposing almost complete graphs by random trees

An old conjecture of Ringel states that every tree with m edges decomposes the complete graph K2m+1. The best known lower bound for the order of a complete graph which admits a decomposition by every given tree with m edges is O(m3). We show that asymptotically almost surely a random tree with m edges and p=2m+1 a prime decomposes K2m+1(r) for every r=2, the graph obtained from the complete graph K2m+1 by replacing each vertex by a coclique of order r. Based on this result we show, among other results, that a random tree with m+1 edges a.a.s. decomposes the compete graph K6m+5 minus one edge.Peer ReviewedPostprint (author's final draft

Optimal packings of bounded degree trees

We prove that if T1,…,Tn is a sequence of bounded degree trees such that Ti has i vertices, then Kn has a decomposition into T1,…,Tn. This shows that the tree packing conjecture of Gyárfás and Lehel from 1976 holds for all bounded degree trees (in fact, we can allow the first o(n) trees to have arbitrary degrees). Similarly, we show that Ringel's conjecture from 1963 holds for all bounded degree trees. We deduce these results from a more general theorem, which yields decompositions of dense quasi-random graphs into suitable families of bounded degree graphs. Our proofs involve Szemerédi's regularity lemma, results on Hamilton decompositions of robust expanders, random walks, iterative absorption as well as a recent blow-up lemma for approximate decompositions

Optimal packings of bounded degree trees

We prove that if T1,…,Tn is a sequence of bounded degree trees such that Ti has i vertices, then Kn has a decomposition into T1,…,Tn. This shows that the tree packing conjecture of Gyárfás and Lehel from 1976 holds for all bounded degree trees (in fact, we can allow the first o(n) trees to have arbitrary degrees). Similarly, we show that Ringel's conjecture from 1963 holds for all bounded degree trees. We deduce these results from a more general theorem, which yields decompositions of dense quasi-random graphs into suitable families of bounded degree graphs. Our proofs involve Szemerédi's regularity lemma, results on Hamilton decompositions of robust expanders, random walks, iterative absorption as well as a recent blow-up lemma for approximate decompositions

Substructures in large graphs

The first problem we address concerns Hamilton cycles. Suppose G is a large digraph in which every vertex has in- and outdegree at least |G|/2. We show that G contains every orientation of a Hamilton cycle except, possibly, the antidirected one. The antidirected case was settled by DeBiasio and Molla. Our result is best possible and improves on an approximate result by Häggkvist and Thomason. We then investigate the random greedy F-free process which was initially studied by Erdős, Suen and Winkler and by Spencer. This process greedily adds edges without creating a copy of F, terminating in a maximal F-free graph. We provide an upper bound on the number of hyperedges at the end of this process for a large class of hypergraphs. The remainder of this thesis focuses on F-decompositions, i.e., whether the edge set of a graph can be partitioned into copies of F. We obtain the best known bounds on the minimum degree which ensures a K$_r$-decomposition of an r-partite graph, with applications to Latin squares. Lastly, we find exact bounds on the minimum degree for a large graph to have a C$_2$$_k$-decomposition where k≠3. In both cases, we assume necessary divisibility conditions are satisfied