In this note, we present a fractional version of Haemers' bound on the Shannon capacity of a graph, which is originally due to Blasiak. This bound is a common strengthening of both Haemers' bound and the fractional chromatic number of a graph. We show that this fractional version outperforms any bound on the Shannon capacity that could be attained through Haemers' bound. We show also that this ...

We study online interval coloring problems with bandwidth. We are interested in some variants motivated by bin packing problems. Specifically we consider open-end coloring, cardinality constrained coloring, coloring with vector constraints and finally a combination of both the cardinality and the vector constraints. We construct competitive algorithms for each of the variants. Additionally, we ...

In this paper the new coloring of planar, VEF-coloring, will be introduced. A VEF coloring of a simple planar graph G is a proper coloring of all elements, including vertices, edges and faces of G. We will give two conjectures for the upper bound of VEF and VEF-list coloring of a simple planar graph. However, we will prove these conjectures for planar graphs with a maximum degree of at least 12...

An ℓ-facial edge-coloring of a plane graph is coloring its edges such that any two at distance most ℓ on boundary walk face receive distinct colors. It the variant vertex coloring, which arose as generalization well-known cyclic coloring. conjectured 3ℓ+1 colors suffice for an graph. The conjecture has only been confirmed ℓ≤2, and in this paper, we prove validity ℓ=3.

pepper belongs to the family of solanaceae that is being used as a food flavoring, a coloring agent, a pharmaceutical ingredient and in other innovative ways. this study was carried out to evaluation of genetic diversity in pepper genotypes using 40 morphological traits on 77 genotypes in augmented design with three controls. measuring of traits of pepper genotypes were performed based on ecpgr...

Determining the sequence of inks in a questioned document is important forensic science. Conventional and micro beam-based ion beam analysis using Rutherford backscattering spectrometry (RBS) particle-induced X-ray emission were employed to study depth distribution chemical elements plain paper inks/toner deposited by different pens as well inkjet laser printers. Composition profiling with high...