In the mid-1990s, Stanley and Stembridge conjectured that the chromatic symmetric functions of claw-free co-comparability (also called incomparability) graphs were $e$-positive. The quest for the proof of this conjecture has led to an examination of other, related graph classes. In 2013 Guay-Paquet proved that if unit interval graphs are $e$-positive, that implies claw-free incomparability graphs are as well. Inspired by this approach, we consider a related case and prove that unit interval graphs whose complement is also a unit interval graph are $e$-positive. We introduce the concept of strongly $e$-positive to denote a graph whose induced subgraphs are all $e$-positive, and conjecture that a graph is strongly $e$-positive if and only if it is (claw, net)-free.
@article{10_37236_8211,
author = {Ang\`ele M. Foley and Ch{\'\i}nh T. Ho\`ang and Owen D. Merkel},
title = {Classes of graphs with \(e\)-positive chromatic symmetric function},
journal = {The electronic journal of combinatorics},
year = {2019},
volume = {26},
number = {3},
doi = {10.37236/8211},
zbl = {1420.05177},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8211/}
}
TY - JOUR
AU - Angèle M. Foley
AU - Chính T. Hoàng
AU - Owen D. Merkel
TI - Classes of graphs with \(e\)-positive chromatic symmetric function
JO - The electronic journal of combinatorics
PY - 2019
VL - 26
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/8211/
DO - 10.37236/8211
ID - 10_37236_8211
ER -
%0 Journal Article
%A Angèle M. Foley
%A Chính T. Hoàng
%A Owen D. Merkel
%T Classes of graphs with \(e\)-positive chromatic symmetric function
%J The electronic journal of combinatorics
%D 2019
%V 26
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/8211/
%R 10.37236/8211
%F 10_37236_8211
Angèle M. Foley; Chính T. Hoàng; Owen D. Merkel. Classes of graphs with \(e\)-positive chromatic symmetric function. The electronic journal of combinatorics, Tome 26 (2019) no. 3. doi: 10.37236/8211
