Adjacent cycle-chains are $e$-positive

TL;DR

This paper introduces a decomposition method for chromatic symmetric functions, demonstrating $e$-positivity for cycle-based graphs and extending results to complex structures formed by cycles and cliques.

Contribution

It presents a novel decomposition approach that proves $e$-positivity for graphs built from cycles and cliques, advancing understanding of chromatic symmetric functions.

Findings

Decomposition of chromatic symmetric functions into positive sums.

$e$-positivity established for cycle-based graphs.

Extension of $e$-positivity results to complex graphs with cycles and cliques.

Abstract

We describe a way to decompose the chromatic symmetric function as a positive sum of smaller pieces. We show that these pieces are $e$-positive for cycles. Then we prove that attaching a cycle to a graph preserves the $e$-positivity of these pieces. From this, we prove an $e$-positive formula for graphs of cycles connected at adjacent vertices. We extend these results to graphs formed by connecting a sequence of cycles and cliques.