Chromatic posets

Dahlberg, Samantha; She, Adrian; Willigenburg, Stephanie van

doi:10.1016/j.jcta.2021.105496

Cited by 2 publications

(1 citation statement)

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“…It may, however, be possible to define such a function implicitly for labelled unit interval graphs at their rightmost vertex by requiring the function satisfy a q-analogue of the relations in Proposition 5.1 and taking the function evaluated at In the first part of [7, Conjecture 6.1], Dahlberg, She and van Willigenburg conjectured that for any connected labelled unit interval graph G, if one constructs a second labelled unit interval graph G ′ by following a certain procedure, then G ≥ e G ′ in the chromatic e-positivity poset. In [7,Remark 6.3], they note that this conjecture would imply the Stanley-Stembridge conjecture. It may be fruitful to also study either a q-analogue of this newer conjecture or a version of it in UBCSym (or a version combining the two, if an appropriate definition for the chromatic quasisymmetric function centred at a vertex exists for labelled unit interval graphs at their rightmost vertex).…”

Section: Ubcqsym and Acyclic Orientationsmentioning

confidence: 99%

The chromatic symmetric function of a graph centred at a vertex

Aliniaeifard¹,

Wang²,

Willigenburg³

2021

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We discover new linear relations between the chromatic symmetric functions of certain sequences of graphs and apply these relations to find new families of e-positive unit interval graphs. Motivated by the results of Gebhard and Sagan, we revisit their ideas and reinterpret their equivalence relation in terms of a new quotient algebra of NCSym. We investigate the projection of the chromatic symmetric function Y G in noncommuting variables in this quotient algebra, which defines y G:v , the chromatic symmetric function of a graph G centred at a vertex v. We then apply our methods to y G:v and find new families of unit interval graphs that are (e)-positive, a stronger condition than classical e-positivity, thus confirming new cases of the (3 + 1)-free conjecture of Stanley and Stembridge.In our study of y G:v , we also describe methods of constructing new e-positive graphs from given (e)-positive graphs and classify the (e)-positivity of trees and cut vertices. We moreover construct a related quotient algebra of NCQSym to prove theorems relating the coefficients of y G:v to acyclic orientations of graphs, including a noncommutative refinement of Stanley's sink theorem.

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Section: Ubcqsym and Acyclic Orientationsmentioning

confidence: 99%