In this paper we design a protocol to extract random bits with an arbitrarily low bias from a single arbitrarily weak min-entropy block source in a device independent setting. The protocol employs Mermin devices that exhibit super-classical correlations. Number of devices used scales polynomially in the length of the block n, containing entropy of at least two bits. Our protocol is robust, it c...

The chromatic number of a graph G, denoted by χ(G), is the minimum number of colors such that G can be colored with these colors in such a way that no two adjacent vertices have the same color. A clique in a graph is a set of mutually adjacent vertices. The maximum size of a clique in a graph G is called the clique number of G. The Turán graph Tn(k) is a complete k-partite graph whose partition...

An edge-colored connected graph G is properly if between every pair of distinct vertices, there exists a path such that no two adjacent edges have the same color. Fujita (2019) introduced optimal proper connection number pc opt (G) for monochromatic , to make efficiently. More precisely, ( ) smallest integer p + q when one converts given into by recoloring with colors. In this paper, we show ha...

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 ...