A necessary condition for four-coloring a planar graph

A Necessary Condition for Weyl-Heisenberg Frames

A practical algorithm for [r, s, t]-coloring of graph

Coloring graphs is one of important and frequently used topics in diverse sciences. In the majority of the articles, it is intended to find a proper bound for vertex coloring, edge coloring or total coloring in the graph. Although it is important to find a proper algorithm for graph coloring, it is hard and time-consuming too. In this paper, a new algorithm for vertex coloring, edge coloring an...

Planar Graph Coloring

We show that the game chromatic number of a planar graph is at most 33. More generally, there exists a function f: f\l --+ f\l so that for each n E f\l. if a graph does not contain a homeomorph of Kn• then its game chromatic number is at most f(n). In particular, the game chromatic number of a graph is bounded in terms of its genus. Our proof is motivated by the concept of p-arrangeability, whi...

Coloring the square of a planar graph

We prove that for any planar graph G with maximum degree , it holds that the chromatic number of the square of G satisfies (G) 2 þ 25. We generalize this result to integer labelings of planar graphs involving constraints on distances one and two in the graph. 2002 Wiley Periodicals, Inc. J Graph Theory 42: 110–124, 2003

A necessary condition for multiple objective fractional programming

In this paper, we establish a proof for  a  necessary condition for  multiple objective fractional programming. In order to derive the set of necessary conditions, we employ an equivalent parametric problem. Also, we  present the related semi parametric model.