An Algebraic Characterization of Rainbow Connectivity

TL;DR

This paper explores algebraic methods to analyze rainbow connectivity in graphs, formulating the problem as polynomial systems and ideal membership problems, extending known results from two colors to the general case.

Contribution

It introduces an algebraic framework for rainbow connectivity, providing polynomial and ideal membership formulations that generalize previous approaches.

Findings

Polynomial systems characterize rainbow connectivity.

Extended algebraic approach to multiple colors.

Formulated rainbow connectivity as an ideal membership problem.

Abstract

The use of algebraic techniques to solve combinatorial problems is studied in this paper. We formulate the rainbow connectivity problem as a system of polynomial equations. We first consider the case of two colors for which the problem is known to be hard and we then extend the approach to the general case. We also give a formulation of the rainbow connectivity problem as an ideal membership problem.