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Let $\vec{T}_k$ be the transitive tournament on $k$ vertices. We show that every oriented graph on $n=4m$ vertices with minimum total degree $(11/12+o(1))n$ can be partitioned into vertex disjoint $\vec{T}_4$'s, and this bound is asymptotically tight. We also improve the best known bound on the minimum total degree for partitioning oriented graphs into vertex disjoint $\vec{T}_k$'s., 18 pages, 3 figures. Minor updates based on referee reports

Corradi and Hajnal (Acta Math Acad Sci Hung 14:423–439, 1963) proved that for all $$k\ge 1$$ and $$n\ge 3k$$ , every (simple) graph G on n vertices with minimum degree $$\delta (G)\ge 2k$$ contains k disjoint cycles. The degree bound is sharp. Enomoto and Wang proved the following Ore-type refinement of the Corradi–Hajnal theorem: For all $$k\ge 1$$ and $$n\ge 3k$$ , every graph G on n vertices contains k disjoint cycles, provided that $$d(x)+d(y)\ge 4k-1$$ for all distinct nonadjacent vertices x, y. Very recently, it was refined for $$k\ge 3$$ and $$n\ge 3k+1$$ : If G is a graph on n vertices such that $$d(x)+d(y)\ge 4k-3$$ for all distinct nonadjacent vertices x, y, then G has k vertex-disjoint cycles if and only if the independence number $$\alpha (G)\le n-2k$$ and G is not one of two small exceptions in the case $$k=3$$ . But the most difficult case, $$n=3k$$ , was not handled. In this case, there are more exceptional graphs, the statement is more sophisticated, and some of the proofs do not work. In this paper we resolve this difficult case and obtain the full picture of extremal graphs for the Ore-type version of the Corradi–Hajnal theorem. Since any k disjoint cycles in a 3k-vertex graph G must be 3-cycles, the existence of such k cycles is equivalent to the existence of an equitable k-coloring of the complement of G. Our proof uses the language of equitable colorings, and our result can be also considered as an Ore-type version of a partial case of the Chen–Lih–Wu Conjecture on equitable colorings.

Corradi and Hajnal in 1963 proved the following theorem on the NP-complete problem on the existence of k disjoint cycles in an n-vertex graph G: For all k ≥ 1 and n ≥ 3k, every (simple) n-vertex graph G with minimum degree δ(G) ≥ 2k contains k disjoint cycles. The same year, Dirac described the 3-connected multigraphs not containing two disjoint cycles and asked the more general question: Which (2k − 1)-connected multigraphs do not contain k disjoint cycles? Recently, Kierstead, Kostochka, and Yeager resolved this question. In this paper, we sharpen this result by presenting a description that can be checked in polynomial time of all multigraphs G with no k disjoint cycles for which the underlying simple graph \( \underline {G}\) satisfies the following Ore-type condition: \(d_{ \underline {G}}(v)+d_{ \underline {G}}(u)\geq 4k-3\) for all nonadjacent u, v ∈ V (G).

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 $\vv G$ with |$\vv G$| = ks and δ($\vv G$) ⩾ 2k(s − 1) − 1 contains k disjoint transitive tournaments on s vertices, where δ($\vv G$)= minv∈V($\vv G$)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.

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 $\delta(G) \ge n/3 + o(n)$ then $G$ has a triangle-tiling covering all but at most four vertices. Also, for every $r \ge 5$, we asymptotically determine the minimum degree threshold for a perfect triangle-tiling under the additional assumptions that $G$ is $K_r$-free and $n$ is divisible by $3$., Comment: 24 pages

In 1963, Corradi and Hajnal settled a conjecture of Erd?s by proving that, for all $$k \ge 1$$k?1, any graph G with $$|G| \ge 3k$$|G|?3k and minimum degree at least 2k contains k vertex-disjoint cycles. In 2008, Finkel proved that for all $$k \ge 1$$k?1, any graph G with $$|G| \ge 4k$$|G|?4k and minimum degree at least 3k contains k vertex-disjoint chorded cycles. Finkel's result was strengthened by Chiba, Fujita, Gao, and Li in 2010, who showed, among other results, that for all $$k \ge 1$$k?1, any graph G with $$|G| \ge 4k$$|G|?4k and minimum Ore-degree at least $$6k-1$$6k-1 contains k vertex-disjoint chorded cycles. We refine this result, characterizing the graphs G with $$|G| \ge 4k$$|G|?4k and minimum Ore-degree at least $$6k-2$$6k-2 that do not have k vertex-disjoint chorded cycles.

Let $n$ be sufficiently large and suppose that $G$ is a digraph on $n$ vertices where every vertex has in- and outdegree at least $n/2$. We show that $G$ contains every orientation of a Hamilton cycle except, possibly, the antidirected one. The antidirected case was settled by DeBiasio and Molla, where the threshold is $n/2+1$. Our result is best possible and improves on an approximate result by H\"aggkvist and Thomason., Comment: Final version, to appear in SIAM Journal Discrete Mathematics (SIDMA)