We study relations between diameter $D(G)$, domination number $\gamma(G)$, independence $\alpha(G)$ and cop $c(G)$ of a connected graph $G$, showing (i.) $c(G) \leq \alpha(G)-\lfloor \frac{D(G)-3}{2} \rfloor$, (ii.) \gamma (G) - \frac{D(G)}{3} + O (\sqrt{D(G)})$.

The linear vertex-arboricity of a graph G is defined to the minimum number of subsets into which the vertex-set G can be partitioned so that every subset induces a linear forest. In this paper, we give the upper and lower bounds for sum and product of linear vertex-arboricity with independence number and with clique cover number respectively. All of these bounds are sharp.

Let s be an integer, f = f(n) a function, and H a graph. Define the Ramsey-Turán number RTs(n,H, f) as the maximum number of edges in an H-free graph G of order n with αs(G) < f , where αs(G) is the maximum number of vertices in a Ks-free induced subgraph of G. The Ramsey-Turán number attracted a considerable amount of attention and has been mainly studied for f not too much smaller than n. In ...

The hypergraph product G2H has vertex set V (G) × V (H), and edge set {e × f : e ∈ E(G), f ∈ E(H)}, where × denotes the usual cartesian product of sets. We construct a hypergraph sequence {Gn} for with χ(Gn) → ∞ and χ(Gn2Gn) = 2 for all n. This disproves a conjecture of Berge and Simonovits [2]. On the other hand, we show that if G and H are hypergraphs with infinite chromatic number, then the ...

The convexity number denoted by in a connected graph is the maximum cardinality of a proper convex set in . Here in this paper graphs for which the independence number ( ) of a graph where ( ) = , ( ) < and ( ) > are completely characterised. Also graphs for which ( ) = are characterised. Construction of graphs with prescribed ( ) and are presented.