Related papers: Chromatic Quasisymmetric Class Functions for combi…

The NP-complete problems Colouring and k-Colouring $(k\geq 3$) are well studied on $H$-free graphs, i.e., graphs that do not contain some fixed graph $H$ as an induced subgraph. We research to what extent the known polynomial-time…

For a hypergraph $\mathcal{H}$, the DP color function $P_{DP}(\mathcal{H},k)$ of $\mathcal{H}$ is an extension of the chromatic polynomial $P(\mathcal{H},k)$ with the property that $P_{DP}(\mathcal{H},k) \le P(\mathcal{H},k)$ for all…

Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We…

This paper investigates methods for calculating the chromatic symmetric function (CSF) of a graph in chromatic-bases and the $m_\lambda$-basis. Our key contributions include a novel approach for calculating the CSF in chromatic-bases…

If we consider previously introduced extensions of Stanley's chromatic symmetric function $X_{G}(x_1, x_2, \ldots)$ for a graph $G$ to elements in the algebra $\textsf{QSym}$ of quasisymmetric functions and in the algebra $\textsf{NCSym}$…

For integer q>1, we derive edge q-colouring models for (i) the Tutte polynomial of a graph G on the hyperbola H_q, (ii) the symmetric weight enumerator of the set of group-valued q-flows of G, and (iii) a more general vertex colouring model…

Independently posed by Behzad and Vizing, the Total Coloring Conjecture asserts that the total chromatic number of a simple connected graph $G$ is either $\Delta(G)+1$ or $\Delta(G)+2$, where $\Delta(G)$ is the largest degree of any vertex…

If G is a graph and H is a set of subgraphs of G, then an edge-coloring of G is called H-polychromatic if every graph from H gets all colors present in G on its edges. The H-polychromatic number of G, denoted poly_H(G), is the largest…

In our previous work, Hopf cyclic cohomology in braided monoidal categories, we extended the formalism of Hopf cyclic cohomology due to Connes and Moscovici and the more general case of Hopf cyclic cohomology with coefficients to the…

In this paper, we study the homology of the cyclic coloring complex of three different types of $k$-uniform hypergraphs. For the case of a complete $k$-uniform hypergraph, we show that the dimension of the $(n-k-1)^{st}$ homology group is…

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 use a Dyck path model for unit-interval graphs to study the chromatic quasisymmetric functions introduced by Shareshian and Wachs, as well as vertical strip --- in particular, unicellular LLT polynomials. We show that there are parallel…

Let $G$ be a finite group and $\chi: G \rightarrow \mathbb{C}$ a class function. Let $H = (V,E)$ be a directed graph with for each vertex a cyclic order of the edges incident to it. The cyclic orders give a collection $F$ of faces of $H$.…

We find families of graphs $G$ and subgraphs $H$ of $G$ such that the number of edge colorings of $G$ avoiding a monochromatic coloring of $H$ is determined by lattice point counts or a Hodge structure on the cohomology of a certain toric…

This paper discusses ways to categorify chromatic, dichromatic and Penrose polynomials, including categorifications of integer evaluations of chromatic polynomials. We show that with an appropriate choice of variables the coefficients of…

We develop a quasisymmetric analogue of the combinatorial theory of Schubert polynomials and the associated divided difference operators. Our counterparts are "forest polynomials", and a new family of linear operators, whose theory of…

We study quasipolynomials enumerating proper colorings, nowhere-zero tensions, and nowhere-zero flows in an arbitrary CW-complex $X$, generalizing the chromatic, tension and flow polynomials of a graph. Our colorings, tensions and flows may…

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 present exact calculations of the partition function of the zero-temperature Potts antiferromagnet (equivalently, the chromatic polynomial) for graphs of arbitrarily great length composed of repeated complete subgraphs $K_b$ with $b=5,6$…

Given a polynomial ring $S = \Bbbk[x_1, \dots, x_n]$ over a field $\Bbbk$, and a monomial ideal $M$ of $S$, we say the quotient ring $R = S/M$ is Macaulay-Lex if for every graded ideal of $R$, there exists a lexicographic ideal of $R$ with…