On the Delta(d)–chromatic number of a complete balanced multipartite graph

Ap Burger,Jh Van Vuuren,I Nieuwoudt

doi:10.5784/23-1-45

Abstract

In this paper we solve (approximately) the problem of finding the minimum number of colours with which the vertices of a complete, balanced, multipartite graph G may be coloured such that the maximum degrees of all colour class induced subgraphs are at most some specified natural number d. The minimum number of colours in such a colouring is referred to as the Delta(d)â€“chromatic number of G. The problem of finding the Delta(d)â€“chromatic number of a complete, balanced, multipartite graph has its roots in an open graph theoretic characterisation problem and has applications conforming to the generic scenario where users of a system are in conflict if they require access to some shared resource. These conflicts are represented by edges in a soâ€“called resource access graph, where vertices represent the users. An efficient resource access schedule is an assignment of the users to a minimum number of groups (modelled by means of colour classes) where some threshold d of conflict may be tolerated in each group. If different colours are associated with different time periods in the schedule, then the minimum number of groupings in an optimal resource access schedule for the above set of users is given by the Delta(d)â€“chromatic number of the resource access graph. A complete balanced multipartite resource access graph represents a situation of maximum conflict between members of different user groups of the system, but where no conflict occurs between members of the same user group (perhaps due to an allocation of diverse duties to the group members).

Highlights

A colouring of a simple graph G is an assignment of colours to the vertices of G, one colour to a vertex, according to some rule R
In this paper we consider a different colouring rule R, namely that the maximum degrees of all colour class induced subgraphs should be bounded from above by some specified integer, and we seek the minimum number of colour classes satisfying this requirement
In this paper we determined a good upper bound on the value of the parameter Dx∆(Kk×n) for the complete balanced multipartite graph Kk×n

Summary

Introduction

A colouring of a simple graph G is an assignment of colours to the vertices of G, one colour to a vertex, according to some rule R. In this paper we consider a different colouring rule R, namely that the maximum degrees of all colour class induced subgraphs should be bounded from above by some specified integer, and we seek the minimum number of colour classes satisfying this requirement We make this definition more precise and outline an application of this colouring problem . Our interest in determining the ∆(d)–chromatic number for the class of complete balanced multipartite graphs stems from a scheduling application where users of a computer system are in conflict if they require access to one or more of the same data files [8].

Normalised k–partite colourings

Concluding remarks