Algorithmes dynamiques pour la communication dans le r\'eseau ad hoc Coloration des graphes

TL;DR

This paper explores advanced graph coloring parameters, especially Grundy numbers, for Cartesian sum graphs, and develops algorithms for graph coloring based on these parameters, with applications to ad hoc network frequency allocation.

Contribution

It introduces new bounds and algorithms for Grundy and partial Grundy coloring parameters in Cartesian sum graphs, extending graph coloring theory.

Findings

Determined bounds for Grundy numbers in Cartesian sum of graphs.

Provided exact values for specific classes of graphs.

Developed algorithms for graph coloring based on Grundy parameters.

Abstract

Several authors modelled networks ad hoc by oriented or disoriented graphs, whereby the problem of allowance (allocation) of the frequencies at the level of the network was transformed into coloring problem of nodes in the graph. Graph coloring is a tool to characterize the graphs. In our study, we were interested in particular in the coloring of vertex. In this domain, a large number of parameters of coloring were defined. A coloring for which two neighboring summits do not have same color is called proper coloring. We proposed to evaluate two other parameters vertex coloring maximizing the number of colors to use: b-chromatic number and especially the number of Grundy. These studies have focused on two types of graphs, which are the powers graphs and Cartesian sum of graphs. In the first part, we determined borders among Grundy for the Cartesian sum of graphs and finally we proposed algorithms for the coloration of graphs. First of all, we gave some properties and results for simple graphs (stable, chain, cycle, full, star, bipartisan). Secondly, we studied the Cartesian sum of two graphs by giving exact values of the number of Grundy for diverse classes of graphs (sum of a chain and a chain, a chain and a cycle, a cycle and a cycle) then we estimated borders of this parameter for the sum of a complete graph and any other graph. In the second part, we presented the results of the Cartesian sum of several graphs and in the last part; we also gave an algorithm for generating graphs with a given number of Grundy. Thereafter, we studied another parameter namely the partial coloring Grundy. The properties of this parameter are close to those of total Grundy. Finally, we used parameters of coloring total Grundy and partial Grundy to generate algorithms of coloring of graphs. We proposed algorithms of coloring basing on the properties of certain graphs.