skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: The Newton polytope and Lorentzian property of chromatic symmetric functions
Chromatic symmetric functions are well-studied symmetric functions in algebraic combinatorics that generalize the chromatic polynomial and are related to Hessenberg varieties and diagonal harmonics. Motivated by the Stanley--Stembridge conjecture, we show that the allowable coloring weights for indifference graphs of Dyck paths are the lattice points of a permutahedron Pλ, and we give a formula for the dominant weight λ. Furthermore, we conjecture that such chromatic symmetric functions are Lorentzian, a property introduced by Brändén and Huh as a bridge between discrete convex analysis and concavity properties in combinatorics, and we prove this conjecture for abelian Dyck paths. We extend our results on the Newton polytope to incomparability graphs of (3+1)-free posets, and we give a number of conjectures and results stemming from our work, including results on the complexity of computing the coefficients and relations with the ζ map from diagonal harmonics.  more » « less
Award ID(s):
2154019
PAR ID:
10543266
Author(s) / Creator(s):
; ;
Publisher / Repository:
Springer
Date Published:
Journal Name:
Selecta Mathematica
Volume:
30
Issue:
3
ISSN:
1022-1824
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; (, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)
    Aichholzer, Oswin; Wang, Haitao (Ed.)
    A graph is said to contain K_k (a clique of size k) as a weak immersion if it has k vertices, pairwise connected by edge-disjoint paths. In 1989, Lescure and Meyniel made the following conjecture related to Hadwiger’s conjecture: Every graph of chromatic number k contains K_k as a weak immersion. We prove this conjecture for graphs with at most 1.4(k-1) vertices. As an application, we make some progress on Albertson’s conjecture on crossing numbers of graphs, according to which every graph G with chromatic number k satisfies cr(G) ≥ cr(K_k). In particular, we show that the conjecture is true for all graphs of chromatic number k, provided that they have at most 1.4(k-1) vertices and k is sufficiently large. 
    more » « less
  2. ; ; ; (, The Electronic Journal of Combinatorics)
    null (Ed.)
    This paper has two main parts. First, we consider the Tutte symmetric function XB, a generalization of the chromatic symmetric function. We introduce a vertex-weighted version of XB and show that this function admits a deletion-contraction relation. We also demonstrate that the vertex-weighted XB admits spanning-tree and spanning-forest expansions generalizing those of the Tutte polynomial by connecting XB to other graph functions. Second, we give several methods for constructing nonisomorphic graphs with equal chromatic and Tutte symmetric functions, and use them to provide specific examples. 
    more » « less
  3. ; ; ; (, Journal of Graph Theory)
    Abstract Let be a multigraph with maximum degree and chromatic index . If is bipartite then . Otherwise, by a theorem of Goldberg, , where denotes the odd girth of . Stiebitz, Scheide, Toft, and Favrholdt in their book conjectured that if then contains as a subgraph a ring graph with the same chromatic index. Vizing's characterization of graphs with chromatic index attaining the Shannon's bound showed the above conjecture holds for . Stiebitz et al verified the conjecture for graphs with and . McDonald proved the conjecture when is divisible by . In this paper, we show that the chromatic index condition alone is not sufficient to give the conclusion in the conjecture. On the positive side, we show that the conjecture holds for every with , and the maximum degree condition is best possible. This positive result leans on the positive resolution of the Goldberg‐Seymour conjecture. 
    more » « less
  4. ; ; ; ; ; ; ; ; ; (, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)
    Felsner, Stefan; Klein, Karsten (Ed.)
    Edge crossings in geometric graphs are sometimes undesirable as they could lead to unwanted situations such as collisions in motion planning and inconsistency in VLSI layout. Short geometric structures such as shortest perfect matchings, shortest spanning trees, shortest spanning paths, and shortest spanning cycles on a given point set are inherently noncrossing. However, the longest such structures need not be noncrossing. In fact, it is intuitive to expect many edge crossings in various geometric graphs that are longest. Recently, Álvarez-Rebollar, Cravioto-Lagos, Marín, Solé-Pi, and Urrutia (Graphs and Combinatorics, 2024) constructed a set of points for which the longest perfect matching is noncrossing. They raised several challenging questions in this direction. In particular, they asked whether the longest spanning path, on any finite set of points in the plane, must have a pair of crossing edges. They also conjectured that the longest spanning cycle must have a pair of crossing edges. In this paper, we give a negative answer to the question and also refute the conjecture. We present a framework for constructing arbitrarily large point sets for which the longest perfect matchings, the longest spanning paths, and the longest spanning cycles are noncrossing. 
    more » « less
  5. ; ; ; (, International Mathematics Research Notices)
    null (Ed.)
    Abstract In this paper, we prove a tight minimum degree condition in general graphs for the existence of paths between two given endpoints whose lengths form a long arithmetic progression with common difference one or two. This allows us to obtain a number of exact and optimal results on cycle lengths in graphs of given minimum degree, connectivity or chromatic number. More precisely, we prove the following statements by a unified approach: 1. Every graph $$G$$ with minimum degree at least $k+1$ contains cycles of all even lengths modulo $$k$$; in addition, if $$G$$ is $$2$$-connected and non-bipartite, then it contains cycles of all lengths modulo $$k$$. 2. For all $$k\geq 3$$, every $$k$$-connected graph contains a cycle of length zero modulo $$k$$. 3. Every $$3$$-connected non-bipartite graph with minimum degree at least $k+1$ contains $$k$$ cycles of consecutive lengths. 4. Every graph with chromatic number at least $k+2$ contains $$k$$ cycles of consecutive lengths. The 1st statement is a conjecture of Thomassen, the 2nd is a conjecture of Dean, the 3rd is a tight answer to a question of Bondy and Vince, and the 4th is a conjecture of Sudakov and Verstraëte. All of the above results are best possible. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email