Abstract. We develop a new method for constructing approximate decompositions of dense graphs into sparse graphs and apply it to longstanding decomposition problems. For instance, our results imply the following. Let G be a quasi-random n-vertex graph and suppose H1, . . . , Hs are bounded degree n-vertex graphs with s i=1 e(Hi) ≤ (1 − o(1))e(G). Then H1, . . . , Hs can be packed edge-disjointly into G. The case when G is the complete graph Kn implies an approximate version of the tree packing conjecture of Gyárfás and Lehel for bounded degree trees, and of the Oberwolfach problem.We provide a more general version of the above approximate decomposition result which can be applied to super-regular graphs and thus can be combined with Szemerédi's regularity lemma. In particular our result can be viewed as an extension of the classical blow-up lemma of Komlós, Sárkőzy and Szemerédi to the setting of approximate decompositions.

No abstract

Abstract:Recently Csikvári [Combinatorica 30(2) 2010, 125-137] proved a conjecture of Nikiforov concerning the number of closed walks on trees. Our aim is to extend this theorem to all walks. In addition, we give a simpler proof of Csikvári's result and answer one of his questions in the negative. Finally we consider an analogous question for paths rather than walks.

No abstract