The $e$-positivity of the chromatic symmetric function for twinned paths and cycles
Abstract
The operation of twinning a graph at a vertex was introduced by Foley, Ho\`ang, and Merkel (2019), who conjectured that twinning preserves -positivity of the chromatic symmetric function. A counterexample to this conjecture was given by Li, Li, Wang, and Yang (2021). In this paper, we prove that -positivity is preserved by the twinning operation on cycles, by giving an -positive generating function for the chromatic symmetric function, as well as an -positive recurrence. We derive similar -positive generating functions and recurrences for twins of paths. Our methods make use of the important triple deletion formulas of Orellana and Scott (2014), as well as new symmetric function identities.
Cite
@article{arxiv.2405.17649,
title = {The $e$-positivity of the chromatic symmetric function for twinned paths and cycles},
author = {Esther Banaian and Kyle Celano and Megan Chang-Lee and Laura Colmenarejo and Owen Goff and Jamie Kimble and Lauren Kimpel and John Lentfer and Jinting Liang and Sheila Sundaram},
journal= {arXiv preprint arXiv:2405.17649},
year = {2025}
}
Comments
34 pages, 2 tables, 9 figures. Minor typos and corrections. We thank J. Zhou for references for the moose graph