Plurigraph Coloring and Scheduling Problems

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 certain scheduling problems. Also, we show that the chromatic symmetric function determines the degree sequence of uniform hypertrees, but there exist pairs of 3-uniform hypertrees which are not isomorphic yet have the same chromatic symmetric function.

New Invariants for Permutations, Orders and Graphs

“…In Section 4 we show that certain properties, including the previously mentioned Schur and e-positivity, are shared by many of the symmetric function invariants coming from the CHAs defined in this paper. We conclude by showing that the invariant we consider can be expressed as scheduling problems as in [12,25].…”

“…It can be advantageous to work with noncommuting variables. For example, there is no deletion-contraction law for the usual chromatic symmetric; however, there is a deletion-contraction law for the chromatic symmetric in noncommuting varaiables [14] as well as for many other scheduling problems [25]. It is thus desirable to know that a problem fits into the context scheduling problems.…”