Abstract
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.
Highlights
A 1995 paper of Stanley [19] introduced the chromatic symmetric functions and proved a host of properties about them
A fundamental contribution to this endeavour was GuayPaquet’s result that if Stanley and Stembridge’s conjecture holds for unit interval graphs, it holds for claw-free co-comparability graphs [8]
Shareshian and Wachs [17] generalized the conjecture of Stanley and Stembridge, introducing an extra t parameter, and conjectured a relation between the chromatic symmetric function and the natural representation of the symmetric group on the cohomology of an algebraic variety
Summary
A 1995 paper of Stanley [19] introduced the chromatic symmetric functions and proved a host of properties about them. As of this writing, this conjecture remains unproved, and work on it and on related results has fueled research. A fundamental contribution to this endeavour was GuayPaquet’s result that if Stanley and Stembridge’s conjecture holds for unit interval graphs, it holds for claw-free co-comparability graphs [8].
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