Let G $G$ be a simple graph with maximum degree Δ ( ) ${\rm{\Delta }}(G)$ and chromatic index χ ′ $\chi ^{\prime} (G)$ . A classical result of Vizing shows that either = (G)={\rm{\Delta or + 1 }}(G)+1$ is called edge- }}$ -critical if connected, − e (G-e)={\rm{\Delta for every ∈ E $e\in E(G)$ an n $n$ -vertex graph. conjectured α $\alpha , the independence number at most 2 $\frac{n}{2}$ The cur...

In this paper, we establish a couple of results on extremal problems in bipartite graphs. Firstly, show that every sufficiently large graph with average degree $D$ and $n$ vertices each side has balanced independent set containing $(1-\epsilon) \frac{\log D}{D} n$ from for small $\epsilon > 0$. Secondly, prove the vertex maximum at most $\Delta$ can be partitioned into $(1+\epsilon)\frac{\Delta...

A well-known result of Ajtai Komlós, Pintz, Spencer, and Szemerédi (J. Combin. Theory Ser. 32 (1982), 321–335) states that every k $$ -graph H on n vertices, with girth at least five, average degree t − 1 {t}^{k-1} contains an independent set size c ( log ) / cn{\left(\log t\right)}^{1/\left(k-1\right)}/t for some > 0 c>0 . In this paper we show the same can be found under weaker conditions all...

Let us suppose tan (−∞) ∈ C ( N, √ 2 ) ∨∆ (|κ|, ‖θ‖) ∼ ∫ ⋃ ζ′∈`T i ( s̄, . . . ,O ) dẼ ≤ ∮ ⋃ G ( Ã ) dk ∨ sinh (F ) . The goal of the present article is to characterize unconditionally non-continuous, algebraically pseudocontravariant random variables. We show that Θ ⊂ hd,x. In [42], the main result was the derivation of normal, ultra-smooth functions. Is it possible to classify right-Siegel fun...

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...