The $e$-positivity of the chromatic symmetric function for twinned paths and cycles

TL;DR

This paper investigates the preservation of $e$-positivity in chromatic symmetric functions under the graph operation of twinning, proving it for cycles and paths using generating functions, recurrences, and symmetric function identities.

Contribution

It proves that $e$-positivity is preserved by twinning on cycles and paths, providing explicit $e$-positive generating functions and recurrences.

Findings

$e$-positivity preserved for cycles under twinning

$e$-positive generating functions derived for twinned paths

New symmetric function identities developed

Abstract

The operation of twinning a graph at a vertex was introduced by Foley, Ho\`ang, and Merkel (2019), who conjectured that twinning preserves $e$-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 $e$-positivity is preserved by the twinning operation on cycles, by giving an $e$-positive generating function for the chromatic symmetric function, as well as an $e$-positive recurrence. We derive similar $e$-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.