The e-positivity of the chromatic symmetric function for twinned paths and cycles
The operation of twinning a graph at a vertex was introduced by Foley, Hoàng, and Merkel (2019), who conjectured that twinning preserves e-positivity of the chromatic symmetric function. A counterexample to this conjecture was given by Li, Li, Wang, and Yang (2021). In this paper, we prove that e-po...
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| Published in | Discrete mathematics Vol. 348; no. 12; p. 114687 |
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| Main Authors | , , , , , , , , , |
| Format | Journal Article |
| Language | English |
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Elsevier B.V
01.12.2025
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| ISSN | 0012-365X |
| DOI | 10.1016/j.disc.2025.114687 |
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| Abstract | The operation of twinning a graph at a vertex was introduced by Foley, Hoàng, and Merkel (2019), who conjectured that twinning preserves e-positivity of the chromatic symmetric function. A counterexample to this conjecture was given by Li, Li, Wang, and Yang (2021). In this paper, we prove that e-positivity is preserved by the twinning operation on cycles, by giving an e-positive generating function for the chromatic symmetric function, as well as an e-positive recurrence. We derive similar e-positive generating functions and recurrences for twins of paths. Our methods make use of the important triple deletion formulas of Orellana and Scott (2014), as well as new symmetric function identities. |
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| AbstractList | The operation of twinning a graph at a vertex was introduced by Foley, Hoàng, and Merkel (2019), who conjectured that twinning preserves e-positivity of the chromatic symmetric function. A counterexample to this conjecture was given by Li, Li, Wang, and Yang (2021). In this paper, we prove that e-positivity is preserved by the twinning operation on cycles, by giving an e-positive generating function for the chromatic symmetric function, as well as an e-positive recurrence. We derive similar e-positive generating functions and recurrences for twins of paths. Our methods make use of the important triple deletion formulas of Orellana and Scott (2014), as well as new symmetric function identities. |
| ArticleNumber | 114687 |
| Author | Celano, Kyle Liang, Jinting Kimble, Jamie Banaian, Esther Colmenarejo, Laura Chang-Lee, Megan Sundaram, Sheila Goff, Owen Lentfer, John Kimpel, Lauren |
| Author_xml | – sequence: 1 givenname: Esther surname: Banaian fullname: Banaian, Esther email: estherb@ucr.edu organization: Department of Mathematics, University of California, Riverside, CA, United States of America – sequence: 2 givenname: Kyle surname: Celano fullname: Celano, Kyle email: celanok@wfu.edu organization: Department of Mathematics, Wake Forest University, NC, United States of America – sequence: 3 givenname: Megan surname: Chang-Lee fullname: Chang-Lee, Megan email: megan_chang-lee@brown.edu organization: Department of Mathematics, Brown University, Providence, RI, United States of America – sequence: 4 givenname: Laura surname: Colmenarejo fullname: Colmenarejo, Laura email: lcolmen@ncsu.edu organization: Department of Mathematics, North Carolina State University, Raleigh, NC, United States of America – sequence: 5 givenname: Owen surname: Goff fullname: Goff, Owen email: owencgoff@gmail.com organization: Department of Mathematics, University of Wisconsin, Madison, WI, United States of America – sequence: 6 givenname: Jamie surname: Kimble fullname: Kimble, Jamie email: jamkimble@gmail.com organization: Department of Mathematics, Michigan State University, East Lansing, MI, United States of America – sequence: 7 givenname: Lauren surname: Kimpel fullname: Kimpel, Lauren email: lkimpel@udel.edu organization: Department of Mathematics, University of Delaware, Newark, DE, United States of America – sequence: 8 givenname: John surname: Lentfer fullname: Lentfer, John email: jlentfer@berkeley.edu organization: Department of Mathematics, University of California, Berkeley, CA, United States of America – sequence: 9 givenname: Jinting surname: Liang fullname: Liang, Jinting email: liangj@math.ubc.ca organization: Department of Mathematics, University of British Columbia, Vancouver, BC, Canada – sequence: 10 givenname: Sheila orcidid: 0000-0002-1583-4740 surname: Sundaram fullname: Sundaram, Sheila email: shsund@umn.edu organization: School of Mathematics, University of Minnesota, Minneapolis, MN, United States of America |
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| Cites_doi | 10.1007/s00373-019-02034-1 10.4171/jems/974 10.1016/0097-3165(93)90048-D 10.11650/tjm/210703 10.1016/j.disc.2013.12.006 10.1137/17M1144805 10.1007/s00373-020-02230-4 10.1007/s10801-022-01175-6 10.1006/aima.1995.1020 10.1023/A:1011258714032 10.1016/0012-365X(92)90378-S 10.1007/BF01170773 10.1016/j.aim.2015.12.018 10.1137/18M1216201 10.1007/s00373-018-1928-2 |
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| Keywords | Elementary symmetric function Chromatic symmetric function Graph twinning e-positivity |
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| SubjectTerms | Chromatic symmetric function e-positivity Elementary symmetric function Graph twinning |
| Title | The e-positivity of the chromatic symmetric function for twinned paths and cycles |
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