Digital Object Identifier (DOI)
The Hajnal–Szemeredi theorem states that for any positive integer ´ r and any multiple n of r, if G is a graph on n vertices and δ(G) > (1 − 1/r)n, then G can be partitioned into n/r vertex-disjoint copies of the complete graph on r vertices. We prove a very general analogue of this result for directed graphs: for any positive integer r with r 6= 3 and any sufficiently large multiple n of r, if G is a directed graph on n vertices and every vertex is incident to at least 2(1 − 1/r)n − 1 directed edges, then G can be partitioned into n/r vertex-disjoint subgraphs of size r each of which contain every tournament on r vertices (the case r = 3 is different and was handled previously). In fact, this result is a consequence of a tiling result for standard multigraphs (that is multigraphs where there are at most two edges between any pair of vertices). A related Turan-type result is also proven.
This work is licensed under a Creative Commons Attribution 4.0 License.
Was this content written or created while at USF?
Citation / Publisher Attribution
Forum of Mathematics, Sigma, v. 6, art. e2
Scholar Commons Citation
Czygrinow, Andrzej; Debiasio, Louis; Molla, Theodore; and Treglown, Andrew, "Tilting Directed Graphs with Tournaments" (2018). Mathematics and Statistics Faculty Publications. 13.