Abstract. We show that provided log 50 n/n ≤ p ≤ 1 − n −1/4 log 9 n we can with high probability find a collection of δ(G)/2 edge-disjoint Hamilton cycles in G ∼ Gn,p, plus an additional edge-disjoint matching of size n/2 if δ(G) is odd. This is clearly optimal and confirms, for the above range of p, a conjecture of Frieze and Krivelevich.
Abstract. Let H be a k-graph on n vertices, with minimum codegree at least n/k + cn for some fixed c > 0. In this paper we construct a polynomial-time algorithm which finds either a perfect matching in H or a certificate that none exists. This essentially solves a problem of Karpiński, Ruciński and Szymańska; Szymańska previously showed that this problem is NP-hard for a minimum codegree of n/k − cn. Our algorithm relies on a theoretical result of independent interest, in which we characterise any such hypergraph with no perfect matching using a family of lattice-based constructions.
Abstract. We show that if pn log n the binomial random graph Gn,p has an approximate Hamilton decomposition. More precisely, we show that in this range Gn,p contains a set of edge-disjoint Hamilton cycles covering almost all of its edges. This is best possible in the sense that the condition that pn log n is necessary.
gave a condition on the minimum degree of a graph G on n vertices that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth o(n), thereby proving a conjecture of Bollobás and Komlós (Combin. Probab. Comput., 1999). We strengthen this result in the case when H is bipartite. Indeed, we give an essentially best-possible condition on the degree sequence of a graph G on n vertices that forces G to contain every bipartite graph H on n vertices of bounded degree and of bandwidth o(n). This also implies an Ore-type result. In fact, we prove a much stronger result where the condition on G is relaxed to a certain robust expansion property. Our result also confirms the bipartite case of a conjecture of Balogh, Kostochka and Treglown concerning the degree sequence of a graph which forces a perfect H-packing.
Degree sequence conditionsDirac's theorem and the Hajnal-Szemerédi theorem are best possible in the sense that the minimum degree conditions in both these results cannot be lowered. However, this does not mean that one cannot strengthen these results. Indeed, Chvátal [9] gave a condition on the degree sequence of a graph which ensures Hamiltonicity: Suppose that the degrees of the graph G are d 1 · · · d n . If n 3 and d i i + 1 or d n−i n − i for all i < n/2 then G is Hamiltonian. Notice that Chvátal's theorem is much stronger than Dirac's theorem since it allows for almost half of the vertices of G to have degree less than n/2.Balogh, Kostochka and Treglown [2] proposed the following two conjectures concerning the degree sequence of a graph which forces a perfect H-packing.
Given η ∈ [0, 1], a colouring C of V (G) is an η-majority colouring if at most ηd + (v) out-neighbours of v have colour C(v), for any v ∈ V (G). We show that every digraph G equipped with an assignment of lists L, each of size at least k, has a 2/k-majority L-colouring. For even k this is best possible, while for odd k the constant 2/k cannot be replaced by any number less than 2/(k + 1). This generalizes a result of Anholcer, Bosek and Grytczuk [1], who proved the cases k = 3 and k = 4 and gave a weaker result for general k.
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