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...

In this article we take up the calculation of the minimum number of colors needed to produce a non-trivial coloring of a knot. This is a knot invariant and we use the torus knots of type (2, n) as our case study. We calculate the minima in some cases. In other cases we estimate upper bounds for these minima leaning on the features of modular arithmetic. We introduce a sequence of transformation...

An efficient mail sorting system is mainly based on an accurate optical recognition of the addresses on the envelopes. However, the localizing of the address block (ABL) should be done before the OCR recognition process. The location step is very crucial as it has a great impact on the global performance of the system. Currently, a good localizing step leads to a better recognition rate. The li...

We have developed the first systematic approach for the design of localized algorithms in wireless embedded ad-hoc sensor networks. The approach has three modular phases: information gathering, system structuring, and optimization mechanisms. The phases can be organized in a number of ways, depending on the underlying operating system and media access control mechanisms. As a driver example for...

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.

A graph G is k-critical if G is not (k − 1)-colorable, but every proper subgraph of G is (k − 1)-colorable. A graph G is k-choosable if G has an L-coloring from every list assignment L with |L(v)| = k for all v, and a graph G is k-list-critical if G is not (k − 1)-choosable, but every proper subgraph of G is (k − 1)-choosable. The problem of bounding (from below) the number of edges in a k-crit...

An efficient mail sorting system is mainly based on an accurate optical recognition of the addresses on the envelopes. However, the localizing of the address block should be done before the OCR recognition process. The location step is very crucial as it has a great impact on the global performance of the system. Actually, a good localizing step leads to a better recognition rate. The limit of ...

the colorful paths and rainbow paths have been considered by severalauthors.a colorful directed path in a digraph $g$ is a directed path with $chi(g)$ vertices whose colors are different. a $v$-colorful directed path is such a directed path, starting from $v$. we prove that for a given $3$-regular triangle-free digraph $g$ determining whether there is a proper $chi(g)$-coloring of $g$such that ...

We show complexity results for some generalizations of the graph coloring problem on two classes of perfect graphs, namely clique trees and unit interval graphs. We deal with the μ-coloring problem (upper bounds for the color on each vertex), the precoloring extension problem (a subset of vertices colored beforehand), and a problem generalizing both of them, the (γ, μ)-coloring problem (lower a...

Graph coloring is a fundamental graph problem that is widely applied in a variety of applications. The aim of graph coloring is to minimize the number of colors used to color the vertices in a graph such that no two incident vertices have the same color. Existing solutions for graph coloring mainly focus on computing a good coloring for a static graph. However, since many real-world graphs are ...