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 Query ID="Q3" Text="Please confirm if the inserted city name in affiliation [3] is correct. Amend if necessary." 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, \(P_4\)-sparse graphs, showing that Tsujie’s result on the e-positivity of claw-free, \(P_4\)-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.
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Acknowledgements
The authors thank Owen Merkel for suggesting the problem e-positivity of generalized nets, and Chính Hoàng for suggesting the problem of e-positivity of claw-free, \(P_4\)-sparse graphs. The authors also thank Leo Tenenbaum for providing code used towards this project. This work was supported by the Canadian Tri-Council Research Support Fund. The author A.M.F. was supported by an NSERC Discovery Grant. This research was conducted at the Fields Institute, Toronto, Canada as part of the 2018 Fields Undergraduate Summer Research Program and was funded by that program.
The authors thank the referees whose helpful suggestions have improved the paper.
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Foley, A.M., Kazdan, J., Kröll, L. et al. Spiders and their Kin: An Investigation of Stanley’s Chromatic Symmetric Function for Spiders and Related Graphs. Graphs and Combinatorics 37, 87–110 (2021). https://doi.org/10.1007/s00373-020-02230-4
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DOI: https://doi.org/10.1007/s00373-020-02230-4