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The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of K2n+1 into edge-disjoint copies of a given 2-factor. We show that this can be achieved for all large n. We actually prove a significantly more general result, which allows for decompositions into more general types of factors. In particular, this also resolves the Hamilton-Waterloo problem for large n. [ABSTRACT FROM AUTHOR]

We study the F -decomposition threshold δ F for a given graph F. Here an F -decomposition of a graph G is a collection of edge-disjoint copies of F in G which together cover every edge of G. (Such an F -decomposition can only exist if G is F -divisible, i.e. if e (F) | e (G) and each vertex degree of G can be expressed as a linear combination of the vertex degrees of F.) The F -decomposition threshold δ F is the smallest value ensuring that an F -divisible graph G on n vertices with δ (G) ≥ (δ F + o (1)) n has an F -decomposition. Our main results imply the following for a given graph F , where δ F ⁎ is the fractional version of δ F and χ : = χ (F) : (i) δ F ≤ max ⁡ { δ F ⁎ , 1 − 1 / (χ + 1) } ; (ii) if χ ≥ 5 , then δ F ∈ { δ F ⁎ , 1 − 1 / χ , 1 − 1 / (χ + 1) } ; (iii) we determine δ F if F is bipartite. In particular, (i) implies that δ K r = δ K r ⁎. Our proof involves further developments of the recent 'iterative' absorbing approach. [ABSTRACT FROM AUTHOR]

Motivated by longstanding conjectures regarding decompositions of graphs into paths and cycles, we prove the following optimal decomposition results for random graphs. Let 0 < p < 1 be constant and let G ∼ G n , p . Let odd ( G ) be the number of odd degree vertices in G . Then a.a.s. the following hold: (i) G can be decomposed into ⌊ Δ ( G ) / 2 ⌋ cycles and a matching of size odd ( G ) / 2 . (ii) G can be decomposed into max ⁡ { odd ( G ) / 2 , ⌈ Δ ( G ) / 2 ⌉ } paths. (iii) G can be decomposed into ⌈ Δ ( G ) / 2 ⌉ linear forests. Each of these bounds is best possible. We actually derive (i)–(iii) from ‘quasirandom’ versions of our results. In that context, we also determine the edge chromatic number of a given dense quasirandom graph of even order. For all these results, our main tool is a result on Hamilton decompositions of robust expanders by Kühn and Osthus. [ABSTRACT FROM AUTHOR]