On a Conjecture of Erdos, Hajnal and Moon

Bollobas, B.

doi:10.2307/2315614

Cited by 28 publications

(47 citation statements)

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“…Alon [12] found the following generalization of Corollary 1.5. at A special case of this (r = 2, al = a2 = 1) was conjectured by ErdSs, Hajnal and Moon [104] and was proved by Bollobgts [45] and Wessel [243]. One can prove …”

Section: Previously the Best Hound Was F(d) < ~(D + 1)! (Zaksmentioning

confidence: 87%

Matchings and covers in hypergraphs

Füredi

1988

Graphs and Combinatorics

165

124

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Abstract. Almost all combinatorial question can be reformulated as either a matching or a covering problem of a hypergraph. In this paper we survey some of the important results. CoversA hypergraph H is an ordered pair (X, ~) where X is a finite set (the set of vertices, or points, or elements) and Jt ~ is a collection of subsets of X (called edges, or members of H). We will often use the notation X = V(H), ~ = E(H). The rank of H is r(H) = max{IE[: E ~ ~}. If every member of Jf has r elements we call it r-uniform, or an r-graph. The 2-uniform hypergraphs are called graphs. In almost all cases we will deal with hypergraphs without multiple edges. 2x and (Xr) denote the family of all subsets (all r-subsets) of X, resp. A set T is called a cover (in other words a transversal or a blocking set) of H if it intersects every edge of H, i.e., T N E ~ ~ for all E ~ ~. The minimum cardinality of the covers is denoted by z(H), and called the covering number of H.

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Section: Previously the Best Hound Was F(d) < ~(D + 1)! (Zaksmentioning

confidence: 87%

Matchings and covers in hypergraphs

Füredi

1988

Graphs and Combinatorics

165

124

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“…For t = s, Conjecture 1.1 trivially follows from the ordered result by Bollobás [3]. The other extreme is also easy to handle.…”

Section: The K 23 Casementioning

confidence: 83%

“…Take any vertex v ′ ∈ U ′ other than u ′ , then adding the edge uv ′ cannot create a K (2,3) , so it must create a K (3,2) , with the 2-vertex class being {u ′ , v ′ }. For any such v ′ , let U v ′ ⊆ U be the 3-class of this K (3,2) , so U v ′ consists of u and two neighbors of v ′ . We count the two edges between v ′ and U v ′ for each v ′ ∈ U ′ , v ′ ̸ = u ′ to get a total of 2n − 2 different edges.…”

Section: Lemma 42mentioning

confidence: 99%

“…Once again, this is seen to be tight by selecting s −1 vertices on each side of the bipartite graph and connecting them to every vertex on the opposite side. In the bipartite setting, one can impose an additional restriction on the problem by ordering the two vertex classes of H and requesting that each missing edge create an H respecting the order: the first class of H lies in the first class of G. For example, let K (s,t) be the complete ''ordered'' s-by-t bipartite graph with s vertices in the first class and t vertices in the second, then a bipartite graph G is K (s,t) -saturated if each missing edge creates a K s,t with the svertex class lying in the first class of G. Indeed, the conjecture of Erdős, Hajnal and Moon was independently confirmed by Wessel [9] and Bollobás [3] a few years later as the special case of the following result: sat(K (n,n) , K (s,t) ) = n 2 −(n−s+1)(n−t +1). This was further generalized in the 80s by Alon [1] to complete k-uniform hypergraphs in a k-partite setting using algebraic tools.…”

Section: Introductionmentioning

confidence: 99%

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Ks,t-saturated bipartite graphs

Gan

Korándi

Sudakov

2015

European Journal of Combinatorics

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a b s t r a c tAn n-by-n bipartite graph is H-saturated if the addition of any missing edge between its two parts creates a new copy of H. In 1964, Erdős, Hajnal and Moon made a conjecture on the minimum number of edges in a K s,s -saturated bipartite graph. This conjecture was proved independently by Wessel and Bollobás in a more general, but ordered, setting: they showed that the minimum number of edges in a K (s,t) -saturated bipartite graph is n 2

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“…The conjecture was proved independently by Bollobás [4] and Wessel [30], while all minimum graphs were characterized in [3,31].…”

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confidence: 99%