Plurigraph coloring and scheduling problems

TL;DR

This paper introduces a new vertex coloring concept that generalizes existing colorings across graphs, hypergraphs, and complexes, and explores its algebraic and combinatorial properties, including applications to scheduling and hypertree analysis.

Contribution

It defines a novel vertex coloring framework that unifies multiple coloring types and connects to symmetric functions and scheduling problems, providing new tools and insights.

Findings

Introduces a generalized vertex coloring concept.

Establishes a deletion-contraction formula for associated symmetric functions.

Shows the chromatic symmetric function's limitations in distinguishing hypertrees.

Abstract

We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes. This coloring also generalizes oriented coloring, acyclic coloring, and star coloring. There is an associated symmetric function in noncommuting variables for which we give a deletion-contraction formula. In the case of graphs this symmetric function in noncommuting variables agrees with the chromatic symmetric function in noncommuting variables of Gebhard and Sagan. Our vertex coloring is a special case of the scheduling problems defined by Breuer and Klivans. We show how the deletion-contraction law can be applied to scheduling problems. Also, we show that the chromatic symmetric function determines the degree sequence of uniform hypertrees, but there exist pairs on $3$-uniform hypertrees which are not isomorphic yet have the same chromatic symmetric function.