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 2 -decomposition, where ≥ 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

“…There have been many recent developments bounding the ‐decomposition threshold, that is, the minimum degree which ensures an ‐decomposition in any large graph satisfying the necessary divisibility conditions. General results on the ‐decomposition threshold establishing a close connection to its fractional counterpart are obtained in and . Moreover, determines the asymptotic decomposition threshold for even cycles and generalises this to arbitrary bipartite graphs.…”

“…General results on the ‐decomposition threshold establishing a close connection to its fractional counterpart are obtained in and . Moreover, determines the asymptotic decomposition threshold for even cycles and generalises this to arbitrary bipartite graphs. The results in and can be combined with bounds for the fractional version of this problem in and to obtain good explicit bounds on the ‐decomposition threshold.…”

“…Moreover, determines the asymptotic decomposition threshold for even cycles and generalises this to arbitrary bipartite graphs. The results in and can be combined with bounds for the fractional version of this problem in and to obtain good explicit bounds on the ‐decomposition threshold. Corresponding results for the multipartite setting (with applications to the completion of partially filled Latin squares) were considered in [3], and .…”

“…Note that any graph which has a ‐decomposition is necessarily ‐divisible. For each with , let us define Barber, Kühn, Lo and Osthus proved asymptotically best possible minimum degree bounds for a graph to have a ‐decomposition.…”

“…We structure the proof into extremal cases where we construct the decompositions directly and nonextremal cases where the iterative absorption approach of and remains effective. Along the way to proving Theorem , we also obtain a bipartite version of Theorem , which is stated as Theorem below.…”