Edge-decompositions of graphs with high minimum degree

Abstract

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,

On the exact decomposition threshold for even cycles

“…Our argument is based on an iterative absorption approach. This method was introduced in [11] and further developed in the context of F -decompositions in [2] and [8]. In our setting, the idea of iterative absorption is as follows.…”

“…Cycles of length four are treated separately since in this case we require a higher minimum degree, namely ≥ | |∕ δ G G ( ) 2 3 − 1. In fact, this minimum degree is sufficient (with room to spare) for (2) and it is only finding an absorber, which causes any difficulty. We are able to show that any graph which does not contain an absorber will, as in the previous case, have a welldefined structure and we find a C 4 -decomposition directly.…”

“…Let F and G be graphs and let η > 0. We say that a collection of edge-disjoint copies of F in G is 2 . In this paper, the graph F will always be bipartite, so we can greedily apply the Erdős-Stone theorem to find an η-approximate Fdecomposition of any large graph G. We say that G is 2-divisible if every vertex in G has even degree.…”

“…Barber, Kühn, Lo and Osthus [2] proved asymptotically best possible minimum degree bounds for a graph to have a C k 2 -decomposition.…”

“…partitioned into cycles of length k. We show that if ≥ | |∕ δ G G ( ) 2 3 − 1, then G has a C 4 -decomposition, and…”