Transversal polynomial of r-fold covers

TL;DR

This paper introduces a transversal polynomial for r-fold graph covers, connecting algebraic combinatorics with graph coloring and the Unique Games Conjecture, and demonstrates its evaluation and divisibility properties.

Contribution

It defines a new polynomial for graph covers, proves a contraction-deletion formula, and establishes a divisibility property related to the cover's parameters.

Findings

The polynomial satisfies a contraction-deletion formula.

The polynomial evaluates to zero modulo r^n at a specific point.

The polynomial generalizes concepts related to graph coloring and algebraic combinatorics.

Abstract

We explore the interplay between algebraic combinatorics and algorithmic problems in graph theory by defining a polynomial with connections to correspondence colouring (also known as DP-colouring), a recent generalization of list-colouring, and the Unique Games Conjecture. Like the chromatic polynomial of a graph, we are able to evaluate this polynomial at a point, despite the complexity of computing this polynomial. We construct a cover of a graph $X$ by blowing up each vertex to a set of $r$ vertices and joining each pair of sets corresponding to adjacent vertices by a matching with $r$ edges. To each cover $Y$ of $X$ we associate a polynomial $\xi(Y,t)$, called the transversal polynomial. The coefficient $t^k$ of $\xi(Y,t)$ is the number of $k$-edge induced subgraphs of $Y$ whose vertex set is a transversal of the set system given by the blown-up vertices. We show that $\xi(Y,t)$ satisfies a contraction-deletion formula, and that if $n=|V_X|$ and the cover has index $r$, then $\xi(Y,-(r-1)) \equiv 0 \mod r^n$.