Vectorial solutions to list multicoloring problems on graphs

TL;DR

This paper introduces algebraic methods to characterize and compute list multicoloring solutions on graphs, addressing problems like channel assignment, on-call adjustments, and recoloring constraints.

Contribution

It provides a novel algebraic framework for list multicoloring, including characterizations of permissible weights and solutions to related optimization problems.

Findings

Algebraic description of permissible weights for list multicoloring

Solution to the channel assignment problem using algebraic methods

Methods to find nearest permissible weights for non-permissible cases

Abstract

For a graph $G$ with a given list assignment $L$ on the vertices, we give an algebraical description of the set of all weights $w$ such that $G$ is $(L,w)$-colorable, called permissible weights. Moreover, for a graph $G$ with a given list $L$ and a given permissible weight $w$, we describe the set of all $(L,w)$-colorings of $G$. By the way, we solve the {\sl channel assignment problem}. Furthermore, we describe the set of solutions to the {\sl on call problem}: when $w$ is not a permissible weight, we find all the nearest permissible weights $w'$. Finally, we give a solution to the non-recoloring problem keeping a given subcoloring.