The Chromatic Symmetric Functions of Trivially Perfect Graphs and Cographs

TL;DR

This paper proves that chromatic symmetric functions uniquely identify trivially perfect graphs and shows that claw-free cographs are part of a known class of e-positive graphs, advancing graph identification methods.

Contribution

It establishes that chromatic symmetric functions distinguish trivially perfect graphs and classifies claw-free cographs as e-positive, extending graph identification techniques.

Findings

Chromatic symmetric functions distinguish trivially perfect graphs.

Claw-free cographs are e-positive graphs.

The method parallels proofs for rooted trees.

Abstract

Richard P. Stanley defined the chromatic symmetric function of a simple graph and has conjectured that every tree is determined by its chromatic symmetric function. Recently, Takahiro Hasebe and the author proved that the order quasisymmetric functions, which are analogs of the chromatic symmetric functions, distinguish rooted trees. In this paper, using a similar method, we prove that the chromatic symmetric functions distinguish trivially perfect graphs. Moreover, we also prove that claw-free cographs, that is, $ \{K_{1,3},P_{4}\} $-free graphs belong to a known class of $ e $-positive graphs.