Related papers: The connection between the chromatic function and …

Stanley and Grinberg introduced the symmetric function associated to digraphs, called the Redei-Berge symmetric function. In [8] is shown that this symmetric function arises from a suitable structure of combinatorial Hopf algebra on…

Recently, Stanley and Grinberg introduced a symmetric function associated to digraphs, called the Redei-Berge symmetric function. This function, however, does not satisfy the deletion-contraction property, which is a very powerful tool for…

Stanley and Grinberg introduced a symmetric function associated with digraphs and named it the Redei-Berge symmetric function. This function arises from a suitable combinatorial Hopf algebra on digraphs, which made it possible to assign the…

In a series of recent talks Richard Stanley introduced a symmetric function associated to digraphs called the Redei-Berge symmetric function. This symmetric function enumerates descent sets of permutations corresponding to digraphs. We show…

We extend the definition of the chromatic symmetric function $X_G$ to include graphs $G$ with a vertex-weight function $w : V(G) \rightarrow \mathbb{N}$. We show how this provides the chromatic symmetric function with a natural…

We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes. This coloring also generalizes oriented coloring, acyclic coloring, and star coloring. There is an associated…

Crew and Spirklt generalize Stanley's chromatic symmetric function to vertex-weighted graphs. One of the primary motivations for extending the chromatic symmetric function to vertex-weighted graphs is the existence of a deletion-contraction…

We define vertex-colourings for edge-partitioned digraphs, which unify the theory of P-partitions and proper vertex-colourings of graphs. We use our vertex-colourings to define generalized chromatic functions, which merge the chromatic…

We introduce a quasisymmetric class function associated with a group acting on a double poset or on a directed graph. The latter is a generalization of the chromatic quasisymmetric function of a digraph introduced by Ellzey, while the…

We focus on two specific generalizations of the chromatic symmetric function: one involving universal graphs and the other concerning vertex-weighted graphs. In this paper, we introduce a unified generalization that incorporates both…

We adapt the classical 3-decomposition of any 2-connected graph to the case of simple graphs (no loops or multiple edges). By analogy with the block-cutpoint tree of a connected graph, we deduce from this decomposition a bicolored tree…

The chromatic polynomial and its generalization, the chromatic symmetric function, are two important graph invariants. Celebrated theorems of Birkhoff, Whitney, and Stanley show how both objects can be expressed in three different ways: as…

Plethysm is a fundamental operation in symmetric function theory, derived directly from its connection with representation theory. However, it does not admit a simple combinatorial interpretation, and finding coefficients of Schur function…

In this note we define L-functions of finite graphs and study the particular case of finite cycles in the spirit of a previous paper that studied spectral zeta functions of graphs. The main result is a suggestive equivalence between an…

Finite graphs that have a common chromatic polynomial have the same number of regular $n$-colorings. A natural question is whether there exists a natural bijection between regular $n$-colorings. We address this question using a functorial…

By establishing relations between operators on compositions, we show that the posets of compositions arising from the right and left Pieri rules for noncommutative Schur functions can each be endowed with both the structure of dual graded…

Let $D=\left( V,A\right) $ be a digraph with $n$ vertices, where each arc $a\in A$ is a pair $\left( u,v\right) $ of two vertices. We study the \emph{Redei--Berge symmetric function} $U_{D}$, defined as the quasisymmetric function% \[ \sum…

Stanley [9] introduced the chromatic symmetric function ${\bf X}_G$ associated to a simple graph $G$ as a generalization of the chromatic polynomial of $G$. In this paper we present a novel technique to write ${\bf X}_G$ as a linear…

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…

We extend the Grundy number and the ochromatic number, parameters on graph colorings, to digraph colorings, we call them {\emph{digrundy number}} and {\emph{diochromatic number}}, respectively. First, we prove that for every digraph the…