Schur and $e$-positivity of trees and cut vertices

Dahlberg, Samantha; She, Adrian; Willigenburg, Stephanie van

doi:10.48550/arxiv.1901.02468

Cited by 4 publications

(8 citation statements)

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“…We pose Conjecture 3.9 for further study. The non-e-positivity of some of the above graphs has been confirmed by Dahlberg, She and van Willigenburg [5]. All results in this paper are checked by using Russell's program [21].…”

Section: Introductionmentioning

confidence: 58%

“…Schur positive graphs include the incomparability graphs of (3+1)free posets, the incomparability graph of the natural unit interval order, and the 2-edge-colorable hyperforests; see [13,20,23]. Non-e-positive graphs include the saltire graphs and triangular tower graphs; see [4,5]. We have not noticed any work concentrated on non-Schur positive graphs yet.…”

Section: Introductionmentioning

confidence: 99%

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Non-Schur-positivity of chromatic symmetric functions

Wang¹,

Wang²

2020

Preprint

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We provide a formula for every Schur coefficient in the chromatic symmetric function of a graph in terms of special rim hook tabloids. This formula is useful in confirming the non-Schur positivity of the chromatic symmetric function of a graph, especially when Stanley's stable partition method does not work. As applications, we determine Schur positive fan graphs and Schur positive complete tripartite graphs. We show that any squid graph obtained by adding n leaves to a common vertex on an m-vertex cycle is not Schur positive if m = 2n − 1, and conjecture that neither are the squid graphs with m = 2n − 1.

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Section: Introductionmentioning

confidence: 58%

Section: Introductionmentioning

confidence: 99%

Non-Schur-positivity of chromatic symmetric functions

Wang¹,

Wang²

2020

Preprint

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“…x κ (v) where the sum ranges over all proper colorings κ of G. Recent research on X G has focused on (among other topics) the Stanley-Stembridge conjecture that the chromatic symmetric function of the incomparability graph of a (3 + 1)-free poset is e-positive [2,4,5,8,11,20], the related conjecture that the chromatic symmetric function of a claw-free graph is s-positive [9,17,18], and the conjecture that X G distinguishes nonisomorphic trees [1,12]. Other results have extended the definition of X G to include quasisymmetric functions [7,23] or noncommuting variables [6,10].…”

Section: Introductionmentioning

confidence: 99%

A deletion–contraction relation for the chromatic symmetric function

Crew

Spirkl

2020

European Journal of Combinatorics

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We extend the definition of the chromatic symmetric function X G to include graphs G with a vertex-weight function w : V (G) → N. We show how this provides the chromatic symmetric function with a natural deletion-contraction relation analogous to that of the chromatic polynomial. Using this relation we derive new properties of the chromatic symmetric function, and we give alternate proofs of many fundamental properties of X G .

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“…If X G is e-positive, we also say that G is e-positive for convenience. Many works have been done towards Stanley's conjecture, see for instance [1,3,4,9,12]. The main objective of this paper is to prove the e-positivity of two families of (claw, 2K 2 )-free graphs.…”

Section: Introductionmentioning

confidence: 99%

The $e$-positivity of two families of $(claw, 2K_2)$-free graphs

Li¹,

Yang²

2019

Preprint

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Motivated by Stanley's conjecture about the e-positivity of claw-free incomparability graphs, Hamel and her collaborators studied the e-positivity of (claw, H)-free graphs, where H is a four-vertex graph. In this paper we establish the e-positivity of generalized pyramid graphs and 2K 2 -free unit interval graphs, which are two important families of (claw, 2K 2 )-free graphs. Hence we affirmatively solve one problem proposed by Hamel, Hoàng and Tuero, and another problem considered by Foley, Hoàng and Merkel.

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Schur and $e$-positivity of trees and cut vertices

Cited by 4 publications

References 0 publications

Non-Schur-positivity of chromatic symmetric functions

Non-Schur-positivity of chromatic symmetric functions

A deletion–contraction relation for the chromatic symmetric function

The $e$-positivity of two families of $(claw, 2K_2)$-free graphs

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