Spiders and their Kin: An Investigation of Stanley's Chromatic Symmetric Function for Spiders and Related Graphs

TL;DR

This paper explores the chromatic symmetric functions of spider-related graphs, demonstrating uniqueness, counterexamples to e-positivity, and analyzing horseshoe crab graphs, thereby advancing understanding of graph invariants and positivity properties.

Contribution

It extends the uniqueness of chromatic symmetric functions to generalized spiders, provides counterexamples to e-positivity conjectures, and analyzes horseshoe crab graphs' positivity properties.

Findings

No two generalized spiders share the same chromatic symmetric function.

Generalized nets, a subclass of generalized spiders, are not e-positive.

Most horseshoe crab graphs are e-positive, with one exception.

Abstract

We study the chromatic symmetric functions of graph classes related to spiders, namely generalized spider graphs (line graphs of spiders), and what we call horseshoe crab graphs. We show that no two generalized spiders have the same chromatic symmetric function, thereby extending the work of Martin, Morin and Wagner. Additionally, we establish that a subclass of generalized spiders, which we call generalized nets, has no e-positive members, providing a more general counterexample to the necessity of the claw-free condition. We use yet another class of generalized spiders to construct a counterexample to a problem involving the $e$-positivity of claw-free, P4-sparse graphs, showing that Tsujie's result on the e-positivity of claw-free, P4-free graphs cannot be extended to graphs in this set. Finally, we investigate the e-positivity of another type of graphs, the horseshoe crab graphs (a class of unit interval graphs), and prove the positivity of all but one of the coefficients. This has close connections to the work of Gebhard and Sagan and Cho and Huh.