Abstract
Graphs and Algorithms We say that a hypergraph H is hamiltonian chain saturated if H does not contain a hamiltonian chain but by adding any new edge we create a hamiltonian chain in H. In this paper we ask about the smallest size of a k-uniform hamiltonian chain saturated hypergraph. We present a construction of a family of k-uniform hamiltonian chain saturated hypergraphs with O(n(k-1/2)) edges.
Highlights
Let H be a k-uniform hypergraph on the vertex set V (H) = {v1, ..., vn} with n ≥ k
The question investigated in this paper is the following: What is the minimum number sat(n, Cn(k)) of edges in a hamiltonian chain saturated k-uniform hypergraphs on n vertices?
On the other hand in [19] a construction is given of n-vertex hamiltonian path saturated k-uniform hypergraphs with k! − 2k k/2 ! k/2 !
Summary
Let H be a k-uniform hypergraph on the vertex set V (H) = {v1, ..., vn} with n ≥ k. Vn) of the vertex set is called a hamiltonian chain and denoted Cn(k), if and only if {vi, vi+1, . The question investigated in this paper is the following: What is the minimum number sat(n, Cn(k)) of edges in a hamiltonian chain saturated k-uniform hypergraphs on n vertices?. Furedi and Tuza [13] obtained sat(n, F ) for some particular hypergraphs F with few edges. Pikhurko [22] proved that sat(n, F ) = O(nk−1) for any fixed hypergraph F (generalizing previous result for graphs by Erdos, Furedi and Tuza [13]). On the other hand in [19] a construction is given of n-vertex hamiltonian path saturated k-uniform hypergraphs with. We notice that the same bound holds for sat(n, Pn(k))
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