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We study the minimum degree necessary to guarantee the existence of perfect and almost-perfect triangle-tilings in an n-vertex graph G with sublinear independence number. In this setting, we show that if δ(G) ≥ n/3 + o(n), then G has a triangle-tiling covering all but at most four vertices. Also, for every r ≥ 5, we asymptotically determine the minimum degree threshold for a perfect triangle-tiling under the additional assumptions that G is Kr-free and n is divisible by 3.
The Hajnal–Szemerédi theorem states that for any positive integer
and any multiple
is a graph on
can be partitioned into
vertex-disjoint copies of the complete graph on
vertices. We prove a very general analogue of this result for directed graphs: for any positive integer
and any sufficiently large multiple
is a directed graph on
vertices and every vertex is incident to at least
directed edges, then
can be partitioned into
vertex-disjoint subgraphs of size
each of which contain every tournament on
vertices (the case
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 Turán-type result is also proven.
Hajnal and Szemerédi proved that every graph G with |G| = ks and δ(G)⩾ k(s − 1) contains k disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every directed graph
| = ks and δ(
) ⩾ 2k(s − 1) − 1 contains k disjoint transitive tournaments on s vertices, where δ(
)d−(v)+d+(v). Our result implies the Hajnal–Szemerédi theorem, and its degree bound is optimal. We also make some conjectures regarding even more general results for multigraphs and partitioning into other tournaments. One of these conjectures is supported by an asymptotic result.
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