Weighted Domination and Colouring in Random Graphs

TL;DR

This paper investigates weighted domination and coloring in random graphs, providing probabilistic conditions for optimal domination and bounds on the weighted chromatic number, especially with large moments and inhomogeneous edge probabilities.

Contribution

It introduces probabilistic iteration methods for weighted domination and extends bounds on the chromatic number to inhomogeneous random graphs with weighted vertices.

Findings

Probabilistic conditions for maximizing weighted domination probability.

Bounds on the weighted chromatic number with large weight moments.

Effective estimates for inhomogeneous random graphs' chromatic number.

Abstract

In the first part of this paper, we consider weighted domination in the case where the vertices of the complete graph on~\(n\) vertices are equipped with independent and identically distributed (i.i.d.) weights. We use the probabilistic iteration to determine sufficient conditions for maximizing the weighted domination probability. In the second part, we study a weighted generalization of the chromatic number and estimate the minimum number of colours needed to satisfy the constraints when the weights themselves are random. We show that the "extra" cost incurred for weighted colouring is small if the weights have sufficiently large moments. We also consider inhomogenous random graphs where the edge probabilities are not necessarily all same and obtain bounds for the chromatic number in terms of its averaged edge probabilities.