Andrzej Zak

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A graph G is called (H; k)-vertex stable if G contains a subgraph isomorphic to H ever after removing any of its k vertices. Q(H; k) denotes the minimum size among the sizes of all (H; k)-vertex stable graphs. In this paper we complete the characterization of (Km,n; 1)-vertex stable graphs with minimum size. Namely, we prove that for m ≥ 2 and n ≥ m + 2,(More)
We say that a hypergraph H is hamiltonian path (cycle) saturated if H does not contain an open (closed) hamiltonian chain but by adding any new edge we create an open (closed) hamiltonian chain in H. In this paper we ask about the smallest size of an r-uniform hamil-tonian path (cycle) saturated hypergraph, mainly for r = 3. We present a construction of a(More)
A graph G is called (H; k)-vertex stable if G contains a subgraph isomorphic to H ever after removing any k of its vertices. By stab(H; k) we denote the minimum size among the sizes of all (H; k)-vertex stable graphs. In this paper we present a first (non-trivial) general lower bound for stab(H; k) with regard to the order, connectivity and minimum degree(More)
A graph G is called (H; k)-vertex stable if G contains a subgraph isomorphic to H even after removing any k of its vertices. By stab(H; k) we denote the minimum size among the sizes of all (H; k)-vertex stable graphs. Given an integer q ≥ 2, we prove that, apart of some small values of k, stab(Kq; k) = (2q − 3)(k + 1). This confirms in the affirmative the(More)
For 1 ≤ < k, an-overlapping cycle is a k-uniform hypergraph in which, for some cyclic vertex ordering, every edge consists of k consecutive vertices and every two consecutive edges share exactly vertices. A k-uniform hypergraph H is-Hamiltonian saturated, 1 ≤ ≤ k − 1, if H does not contain an-overlapping Hamiltonian cycle C (k) n () but every hypergraph(More)
Let G be a graph of order n. We prove that if the size of G is less than or equal to n − 2(k − 1) 3 then the complete graph Kn contains k edge-disjoint copies of G. The case when k = 2 is the well known theorem of Sauer and Spencer 1978.