Abstract. A fundamental theorem of Wilson states that, for every graph F , every sufficiently large F -divisible clique has an F -decomposition. Here a graph G is F -divisible if e(F ) divides e(G) and the greatest common divisor of the degrees of F divides the greatest common divisor of the degrees of G, and G has an F -decomposition if the edges of G can be covered by edge-disjoint copies of F . We extend this result to graphs G which are allowed to be far from complete. In particular, together with a result of Dross, our results imply that every sufficiently large K3-divisible graph of minimum degree at least 9n/10 + o(n) has a K3-decomposition. This significantly improves previous results towards the long-standing conjecture of Nash-Williams that every sufficiently large K3-divisible graph with minimum degree at least 3n/4 has a K3-decomposition. We also obtain the asymptotically correct minimum degree thresholds of 2n/3 + o(n) for the existence of a C4-decomposition, and of n/2 + o(n) for the existence of a C 2ℓ -decomposition, where ℓ ≥ 3. Our main contribution is a general 'iterative absorption' method which turns an approximate or fractional decomposition into an exact one. In particular, our results imply that in order to prove an asymptotic version of Nash-Williams' conjecture, it suffices to show that every K3-divisible graph with minimum degree at least 3n/4 + o(n) has an approximate K3-decomposition,
Given integers n ≥ k > l ≥ 1 and a k-graph F with |V (F )| divisible by n, define t k l (n, F ) to be the smallest integer d such that every kgraph H of order n with minimum l-degree δ l (H) ≥ d contains an F -factor. A classical theorem of Hajnal and Szemerédi [9] implies that t 2 1 (n, K t ) = (1 − 1/t)n for integers t. For k ≥ 3, t k k−1 (n, K k k ) (the δ k−1 (H) threshold for perfect matchings) has been determined by Kühn and Osthus [17] (asymptotically) and Rödl, Ruciński and Szemerédi [24] (exactly) for large n.In this paper, we generalise the absorption technique of Rödl, Ruciński and Szemerédi [24] to F -factors. We determine the asymptotic values of t k 1 (n, K k k (m)) for k = 3, 4 and m ≥ 1. In addition, we show that for t > k = 3 and γ > 0, t 3 2 (n, K 3 t ) ≤ 1 − 2 * Lemma 1.1 (Absorption lemma for F -factors). Let t and i be positive integers and let η > 0. Let F be a hypergraph of order t. Then, there is an integer n 0 = n 0 (t, i, η) satisfying the following: Suppose that H is an (F, i, η)-closed hypergraph of order n ≥ n 0 . Then there exists a vertex subset U ⊆ V (H) of size |U | ≤ (η/2) t n/(4it(t − 1)) with |U | ∈ tZ such that there exists an F -factor in H[U ∪ W ] for every vertex set W ⊆ V \ U of size |W | ≤ (η/2) 2t n/(32i 2 t(t − 1) 2 ) with |W | ∈ tZ.Note that in the above lemma H and F are not necessarily kgraphs, but we only consider k-graphs in this paper. When we say that H has an almost F -factor T , we mean that T is a set of vertexdisjoint copies of F in H such that |V (H) \ V (T )| < ε|V (H)| for some small ε > 0. Equipped with the absorption lemma, we can break down the task of finding an F -factor in large hypergraphs H into the following algorithm.Algorithm for finding F -factors. Remove a small set T 1 of vertex-disjoint copies of F from H such that the resultant graph Hfor some integer i and constant η > 0. 2. Find a vertex set U ⊆ V (H 1 ) satisfying the conditions of the absorption lemma. Set H 2 = H 1 [V (H 1 ) \ U ]. 3. Show that H 2 contains an almost F -factor, i.e. a set T 2 of vertexdisjoint copies of F such that |V (H 2 ) \ V (T 2 )| < ε|V (H 2 )| for small ε > 0.
The iterative absorption method has recently led to major progress in the area of (hyper-)graph decompositions. Amongst other results, a new proof of the Existence conjecture for combinatorial designs, and some generalizations, was obtained. Here, we illustrate the method by investigating triangle decompositions: we give a simple proof that a triangle-divisible graph of large minimum degree has a triangle decomposition and prove a similar result for quasirandom host graphs.
In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large n:(i) [1-factorization conjecture] Suppose that n is even and D ≥ 2⌈n/4⌉ − 1. Then every D-regular graph G on n vertices has a decomposition into perfect matchings. Equivalently, χ ′ (G) = D. (ii) [Hamilton decomposition conjecture] Suppose that D ≥ ⌊n/2⌋. Then every D-regular graph G on n vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) We prove an optimal result on the number of edge-disjoint Hamilton cycles in a graph of given minimum degree. According to Dirac, (i) was first raised in the 1950s. (ii) and (iii) answer questions of Nash-Williams from 1970. The above bounds are best possible. In the current paper, we show the following: suppose that G is close to a complete balanced bipartite graph or to the union of two cliques of equal size. If we are given a suitable set of path systems which cover a set of 'exceptional' vertices and edges of G, then we can extend these path systems into an approximate decomposition of G into Hamilton cycles (or perfect matchings if appropriate).
Our main result is that every graph G on n ≥ 10 4 r 3 vertices with minimum degree δ(G) ≥ (1−1/10 4 r 3/2 )n has a fractional Kr-decomposition. Combining this result with recent work of Barber, Kühn, Lo and Osthus leads to the best known minimum degree thresholds for exact (non-fractional) F -decompositions for a wide class of graphs F (including large cliques). For general k-uniform hypergraphs, we give a short argument which shows that there exists a constant c k > 0 such that every k-uniform hypergraph G on n vertices with minimum codegree at least (1 − c k /r 2k−1 )n has a fractional Kis the complete k-uniform hypergraph on r vertices. (Related fractional decomposition results for triangles have been obtained by Dross and for hypergraph cliques by Dukes as well as Yuster.) All the above new results involve purely combinatorial arguments. In particular, this yields a combinatorial proof of Wilson's theorem that every large F -divisible complete graph has an F -decomposition.
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