Andrzej Zak

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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) 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.
Assessments of trust in intimate relationships are often based on perceptions of a partner's behaviors; however, people's own actions, increased self-awareness, and individual differences (e.g., exchange or communal orientation) may also affect their trust in their partners. Communally or exchange-oriented members of heterosexual dating couples, students in(More)
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)
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; stab(H; k) denotes the minimum size among the sizes of all (H; k)-vertex stable graphs. In this paper we deal with (Cn; k)vertex stable graphs with minimum size. For each n we prove that stab(Cn; 1) is one of only two possible values(More)
The following statement was conjectured by Faudree, Rousseau, Schelp and Schuster: if a graph G is a non-star graph without cycles of length m 6 4 then G is a subgraph of its complement. So far the best result concerning this conjecture is that every non-star graph G without cycles of length m 6 6 is a subgraph of its complement. In this note we show that m(More)
One of the classical results in packing theory states that every graph of order n and size less than or equal to n− 2 is packable in its complement. Moreover, the bound is sharp because the star is not packable. A similar problem arises for digraphs, namely, to find the maximal number fD(n) such that every digraph of order n and size less than or equal to(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)