Abstract

We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, andsimplicial complexes. To this coloring there is an associated symmetric function in noncommuting variables for whichwe give a deletion-contraction formula. In the case of graphs our symmetric function in noncommuting variablesagrees with the chromatic symmetric function in noncommuting variables of Gebhard and Sagan. Our vertex coloringis a special case of the scheduling problems defined by Breuer and Klivans. We show how the deletion-contractionlaw can be applied to scheduling problems.

Highlights

  • In this paper we define a generalization of vertex coloring which has graph coloring and hypergraph coloring as a special case

  • Associated to our generalization vertex coloring we have a symmetric function in noncommuting variables which generalizes the chromatic symmetric function in noncommuting variables defined by Gebhard and Sagan in [GS01]

  • The vertex coloring we study corresponds to a special class of the scheduling problems defined by Breuer and Klivans in [BK14], and our symmetric function in noncommuting variables is an instance of the scheduling quasisymmetric function in noncommuting variables from [BK14]

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Summary

Introduction

In this paper we define a generalization of vertex coloring which has graph coloring and hypergraph coloring as a special case. Associated to our generalization vertex coloring we have a symmetric function in noncommuting variables which generalizes the chromatic symmetric function in noncommuting variables defined by Gebhard and Sagan in [GS01]. The vertex coloring we study corresponds to a special class of the scheduling problems defined by Breuer and Klivans in [BK14], and our symmetric function in noncommuting variables is an instance of the scheduling quasisymmetric function in noncommuting variables from [BK14]

NCSym and NCQSym
Scheduling Problems
Coloring and Generalized Graphs
Graph-like Scheduling Problems
Coloring Uniform Hypergraphs and Simplicial Complexes
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