A few more trees the chromatic symmetric function can distinguish

Huryn, Jake; Chmutov, Sergei

doi:10.2140/involve.2020.13.109

Cited by 9 publications

(3 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This conjecture is verified for trees with up to 29 vertices by Heil and Ji [22]. For recent studies, see [1,2,23,24,26,28,31] There is another conjecture concerning the e-positivity of chromatic symmetric functions. See [4,6,7,8,10,11,12,13,15,17,18,19,20,30,36] for recent studies.…”

mentioning

confidence: 86%

Chromatic Signed-Symmetric Functions of Signed Graphs

Kuroda,

Tsujie

2021

Preprint

View full text Add to dashboard Cite

Stanley introduced the chromatic symmetric function of a simple graph, which is a generalization of a chromatic polynomial. This is expressed in terms of the integer points of the complements of the corresponding graphic arrangement. Stanley proved a combinatorial reciprocity theorem for chromatic functions. This is considered as an Ehrhart-type reciprocity theorem for the graphic arrangement.We introduce the chromatic signed-symmetric function of a signed graph, an analogue of the chromatic symmetric function, by the integer points of the complements of the corresponding signed-graphic arrangement and prove a generalization of Stanley's reciprocity theorem.Stanley has conjectured that the chromatic symmetric function distinguishes trees. This conjecture is also generalized for signed trees. We verify the conjecture for certain classes of signed paths.

show abstract

mentioning

confidence: 86%

Chromatic Signed-Symmetric Functions of Signed Graphs

Kuroda,

Tsujie

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Stanley stated "We do not know whether X G distinguishes trees." Subsequent papers, such as [2,3,4,10,14,15,19,22,24,26], have established that X G (x) distinguishing trees is certainly worthy of being called a conjecture; for example, [14] shows that X G (x) distinguishes trees with up to 29 vertices. We focus on a generalization of X G (x) to labeled graphs introduced by Shareshian and Wachs [25], denoted X G (x, t), which has an extra parameter t and is now just a quasisymmetric function in general.…”

Section: • • •mentioning

confidence: 99%

Quasisymmetric functions distinguishing trees

Aval¹,

Djenabou²,

McNamara³

2023

Algebraic Combinatorics

View full text Add to dashboard Cite

A famous conjecture of Stanley states that his chromatic symmetric function distinguishes trees. As a quasisymmetric analogue, we conjecture that the chromatic quasisymmetric function of Shareshian and Wachs and of Ellzey distinguishes directed trees. This latter conjecture would be implied by an affirmative answer to a question of Hasebe and Tsujie about the P -partition enumerator distinguishing posets whose Hasse diagrams are trees. They proved the case of rooted trees and our results include a generalization of their result.

show abstract

“…Using computer investigations, Heil and Ji [22] verified that the conjecture is true for all trees with at most 29 vertices. The conjecture also holds for certain restricted classes of trees: spiders [26], 2-spiders [23], caterpillars [5,24,27], trivially perfect graphs [34], and trees with diameter at most five [3]; see also the families described in [2,Cor. 8.7].…”

mentioning

confidence: 99%

Combinatorics of multivariate chromatic polynomials for rooted graphs

Loehr¹,

Warrington²

2022

Preprint

View full text Add to dashboard Cite

Richard Stanley defined the chromatic symmetric function XG of a graph G and conjectured that trees T and U are isomorphic if and only if XT = XU . We study a variation of the chromatic symmetric function for rooted graphs, where we require the root vertex to have a specified color. In addition to presenting various combinatorial identities and recursions satisfied by these rooted chromatic polynomials, this work contains three main results. The first is that our polynomials satisfy the analogue of Stanley's conjecture: two rooted trees are isomorphic as rooted graphs if and only if their rooted chromatic polynomials are equal. We give two proofs of this theorem, one based on unique factorization, the other based on algebraic transformations to pointed chromatic functions and rooted U -polynomials. The second result, which is used in the factorization-based proof, is that for each tree T , XT (x1, . . . , xN ) is irreducible in Q[x1, . . . , xN ] for large enough N . We prove this by a novel application of Eisenstein's Criterion. The third result answers a question of Pawlowski by providing a combinatorial interpretation of the monomial expansion of pointed chromatic functions.

show abstract

A few more trees the chromatic symmetric function can distinguish

Cited by 9 publications

References 5 publications

Chromatic Signed-Symmetric Functions of Signed Graphs

Chromatic Signed-Symmetric Functions of Signed Graphs

Quasisymmetric functions distinguishing trees

Combinatorics of multivariate chromatic polynomials for rooted graphs

Contact Info

Product

Resources

About