Let H0 be any fixed graph. For a graph G we define νH 0 (G) to be the maximum size of a set of pairwise edge-disjoint copies of H0 in G. We say a function ψ from the set of copiesLemma 4. Let λ > 0 and a graph H 0 be given. Then there exists K = K(λ, H 0 ) such that the following holds. Let G be a graph with n vertices, and let ψ * be a fractional H 0 -packing of G. Then there exists a partition
Moreover there is a O(n 2 ) algorithm which finds such a partition.We say that a pair (A, B) of disjoint vertex subsets of a graph G isregular with density α ± if |d G (A ,B ) − α| < for every A ⊂ A and B ⊂ B with |A |≥ |A| and |B |≥ |B|.
Let k be a positive integer and let G be a graph. Suppose a list S(v)
of positive integers is assigned to each vertex v, such that(1) [mid ]S(v)[mid ] = 2k for each vertex v of G, and(2) for each vertex v, and each c ∈ S(v), the
number of neighbours w of v for which c ∈ S(w) is at most k.Then we prove that there exists a proper vertex colouring f of G such that
f(v) ∈ S(v) for each v ∈ V(G).
This proves a weak version of a conjecture of Reed.
Abstract. An extension of Hall's theorem is proved, which gives a condition for complete matchings in a certain class of hypergraphs.Let ~ = (V, ¢) be a hypergraph. A matching in ~'~ is a set of pairwise disjoint edges, and a transversal is a subset T _ V with the property that E N T ~ ~ for every E ~ ~. We denote by v(Xf) the maximum size of a matching in ,~ff, and by T(~Vf) the minimum size of a transversal of ~. In this note we shall be concerned with hypergraphs ~ = (V, ~), where r > 2 is a given integer and V is a finite set, which satisfy the following property.(*) The ground set V is the disjoint union of sets A and X, and every E e 8 satisfies lENA[ = 1 and lENS[ _< r -1.The main result of this note gives a condition on a hypergraph satisfying (*) which ensures that it has a complete matching from A to X, that is, v(~) = IAI.Several criteria for matchability in hypergraphs ~ satisfying (*) were proposed by Aharoni
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