Andrzej Żak

On vertex stability with regard to complete bipartite subgraphs

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, Q(Km;n; 1) = mn+m+n and Km;n K1 as well as Km+1;n+1 e are the only (Km;n; 1)-vertex stable graphs with minimum size, conrming the conjecture of Dudek and Zwonek.

Hamilton-chain saturated hypergraphs

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 hamiltonian path (cycle) saturated hypergraph, mainly for r=3. We present a construction of a family of 3-uniform path (cycle) saturated hamiltonian hypergraphs with O(n^5^/^2) edges. On the other hand we prove that the number of edges in an r-uniform hamiltonian path (cycle) saturated hypergraph is at least @W(n^r^-^1).

A lower bound on the size of (H;1)-vertex stable graphs

Abstract 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. In this paper we present a general result concerning ( H ; 1 ) -vertex stable graphs. Namely, for an arbitrary graph H we give a lower bound for stab ( H ; 1 ) in terms of the order, connectivity and minimum degree of H . The bound is nearly sharp.

General lower bound on the size of (H;k)-stable graphs

A graph

Sparse graphs of girth at least five are packable

G

On embedding graphs with bounded sum of size and maximum degree

is called

On (Kq;k)-Stable Graphs

(H;k)