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- W4288093303 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 $xi(Y,-(r-1)) equiv 0 mod r^n$." @default.
- W4288093303 created "2022-07-28" @default.
- W4288093303 creator A5071503218 @default.
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- W4288093303 date "2019-10-11" @default.
- W4288093303 modified "2023-09-26" @default.
- W4288093303 title "Transversal polynomial of r-fold covers" @default.
- W4288093303 doi "https://doi.org/10.48550/arxiv.1910.05478" @default.
- W4288093303 hasPublicationYear "2019" @default.
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