The classical Corrádi‐Hajnal theorem claims that every n‐vertex graph G with δ(G)≥2n/3 contains a triangle factor, when 3|n. In this paper we present two related results that both use the absorbing technique of Rödl, Ruciński and Szemerédi. Our main result determines the minimum degree condition necessary to guarantee a triangle factor in graphs with sublinear independence number. In particular, we show that if G is an n‐vertex graph with α(G)=o(n) and δ(G)≥(1/2+o(1))n, then G has a triangle factor and this is asymptotically best possible. Furthermore, it is shown for every r that if every linear size vertex set of a graph G spans quadratically many edges, and δ(G)≥(1/2+o(1))n, then G has a Kr‐factor for n sufficiently large. We also propose many related open problems whose solutions could show a relationship with Ramsey‐Turán theory. Additionally, we also consider a fractional variant of the Corrádi‐Hajnal Theorem, settling a conjecture of Balogh‐Kemkes‐Lee‐Young. Let t∈(0,1) and w:E(Kn)→[0,1]. We call a triangle t‐heavy if the sum of the weights on its edges is more than 3t. We prove that if 3|n and w is such that for every vertex v the sum of w(e) over edges e incident to v is at least (1+2t3+o(1))n, then there are n/3 vertex disjoint heavy triangles in G. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 669–693, 2016
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 K r -free and n is divisible by 3. IntroductionA triangle-tiling in a graph G is a collection T of vertex-disjoint triangles in G. We say that T is perfect if |T | = n/3, where n is the order of G. A trivial necessary condition for the existence of a perfect triangle-tiling is that 3 divides n. We let V (T ) := T ∈T V (T ) and say T covers U ⊆ V (G) (respectively v ∈ V (G)) when U ⊆ V (T ) (respectively v ∈ V (T )), so a perfect triangle-tiling covers every vertex of the host graph. Given disjoint sets A and B which partition V (G), we say that a triangle T in G is an Atriangle if T contains two vertices of A and one vertex of B, and likewise that T is a B-triangle if T contains two vertices of B and one vertex of A. Observe that if |A| = 1 (mod 3) and |B| = 2 (mod 3), there are no B-triangles in G and also there is no pair of vertex-disjoint A-triangles in G, then G does not have a perfect triangle-tiling. In that case, we call the ordered pair (A, B) a divisibility barrier in G (note that order is
Abstract. 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 graphOur 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.
Osthus [J. Combin. Theory Ser. B, 96 (2006), pp. 767-821] showed that if a 3-graph H on n vertices has minimum codegree at least (1/4 + o(1))n and n is even, then H has a loose Hamilton cycle. In this paper, we prove that the minimum codegree of n/4 suffices. The result is tight. Introduction.is a finite set of vertices and the edge set E is a subset of V k . Often, we will identify H with its edges and, if needed, will use V (H) and E(H) to denote the vertex set and the edge set of H, respectively. In this paper we will only use 3-uniform hypergraphs (3-graphs) and 2-uniform hypergraphs (graphs). We say that a 3-graph H is a (loose) path if its vertices can be ordered asH is called a (loose) cycle if its vertices can be ordered as v 1 , v 2 , . . . , v 2m so that
We prove two results regarding cycles in properly edge-colored graphs. First, we make a small improvement to the recent breakthrough work of Alon, Pokrovskiy and Sudakov who showed that every properly edge-colored complete graph G on n vertices has a rainbow cycle on at least n − O(n 3/4 ) vertices, by showing that G has a rainbow cycle on at least n − O(log n √ n) vertices. Second, by modifying the argument of Hatami and Shor which gives a lower bound for the length of a partial transversal in a Latin Square, we prove that every properly colored complete graph has a Hamilton cycle in which at least n − O((log n) 2 ) different colors appear. For large n, this is an improvement of the previous best known lower bound of n − √ 2n of Andersen.
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