Related papers: Bivariate Order Polynomials

Recently, Alon, Cambie, and Kang introduced asymmetric list coloring of bipartite graphs, where the size of each vertex's list depends on its part. For complete bipartite graphs, we fix the list sizes of one part and consider the resulting…

Associated with every graph $G$ of chromatic number $\chi$ is another graph $G'$. The vertex set of $G'$ consists of all $\chi$-colorings of $G$, and two $\chi$-colorings are adjacent when they differ on exactly one vertex. According to a…

We introduce $H$-chromatic symmetric functions, $X_{G}^{H}$, which use the $H$-coloring of a graph $G$ to define a generalization of Stanley's chromatic symmetric functions. We say two graphs $G_1$ and $G_2$ are $H$-chromatically equivalent…

This paper serves as the first extension of the topic of dominator colorings of graphs to the setting of digraphs. We establish the dominator chromatic number over all possible orientations of paths and cycles. In this endeavor we discover…

We establish a set of recursion relations for the coefficients in the chromatic polynomial of a graph or a hypergraph. As an application we provide a generalization of Whitney's broken cycle theorem for hypergraphs, as well as deriving an…

The U-polynomial, the polychromate and the symmetric function generalization of the Tutte polynomial due to Stanley are known to be equivalent in the sense that the coefficients of any one of them can be obtained as a function of the…

A class of graphs is $\chi$-bounded if there is a function $f$ such that every graph $G$ in the class has chromatic number at most $f(\omega(G))$, where $\omega(G)$ is the clique number of $G$; the class is polynomially $\chi$-bounded if…

Stanley introduces polynomials which help evaluate symmetric group characters and conjectures that the coefficients of the polynomials are positive. Stanley later gives a conjectured combinatorial interpretation for the coefficients of the…

Let $\G$ denote a bipartite distance-regular graph with vertex set $X$ and diameter $D \ge 3$. Fix $x \in X$ and let $L$ (resp. $R$) denote the corresponding lowering (resp. raising) matrix. We show that each $Q$-polynomial structure for…

We show that the number of $k$-matching in a given undirected graph $G$ is equal to the number of perfect matching of the corresponding graph $G_k$ on an even number of vertices divided by a suitable factor. If $G$ is bipartite then one can…

The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. Let $\chi(H)$ and $\chi_l(H)$ be the chromatic number and the…

The Tutte polynomial is a fundamental invariant of graphs. In this article, we define and study a generalization of the Tutte polynomial for directed graphs, that we name B-polynomial. The B-polynomial has three variables, but when…

Given a graph $G=(V,E)$ and a linear form $\lambda \in \mathbb{Z}_{ > 0 }^V$, Bajo et al. (2025) introduced the $q$-chromatic polynomial $\chi_G^\lambda(q,n) := \sum q^{\sum_{v \in V} \lambda_v c(v)}$ where the sum is over all proper…

We prove that for any graph $G$, the total chromatic number of $G$ is at most $\Delta(G)+2\left\lceil \frac{|V(G)|}{\Delta(G)+1} \right\rceil$. This saves one color in comparison with a result of Hind from 1992. In particular, our result…

A common generalization for the chromatic polynomial and the flow polynomial of a graph $G$ is the Tutte polynomial $T(G;x,y)$. The combinatorial meaning for the coefficients of $T$ was discovered by Tutte at the beginning of its…

Let $T$ be an unrooted tree. The \emph{chromatic symmetric function} $X_T$, introduced by Stanley, is a sum of monomial symmetric functions corresponding to proper colorings of $T$. The \emph{subtree polynomial} $S_T$, first considered…

We present results on partitioning the vertices of $2$-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of S\'ark\"ozy: the vertex set of every $2$-edge-colored graph can be…

Stanley introduced the concept of chromatic symmetric functions of graphs which extends and refines the notion of chromatic polynomials of graphs, and asked whether trees are determined up to isomorphism by their chromatic symmetric…

We find a combinatorial setting for the coefficients of the Boros-Moll polynomials $P_m(a)$ in terms of partially 2-colored permutations. Using this model, we give a combinatorial proof of a recurrence relation on the coefficients of…

We consider colored compositions where only some parts are allowed different colors, depending on their locations in the composition. The counting sequences are obtained through generating functions. Connections to many other combinatorial…