Related papers: Polynomials Counting Group Colorings in Graphs

The problem of finding the minimum number of colors to color a graph properly without containing any bicolored copy of a fixed family of subgraphs has been widely studied. Most well-known examples are star coloring and acyclic coloring of…

In this note we introduce a family of polynomials on a matroid derived from chain Tutte polynomials which generalize the classic and ubiquitous characteristic polynomial. We show that the coefficients of these polynomials alternate and…

Let $G$ be a graph whose each component has order at least 3. Let $s : E(G) \rightarrow \mathbb{Z}_k$ for some integer $k\geq 2$ be an improper edge coloring of $G$ (where adjacent edges may be assigned the same color). If the induced…

Given an $r$-edge-colouring of the edges of a graph $G$, we say that it can be partitioned into $p$ monochromatic cycles when there exists a set of $p$ vertex-disjoint monochromatic cycles covering all the vertices of $G$. In the literature…

Around 10 years ago, Agol and Krushkal showed that the number of chromatic polynomials $P_{G}$ arising from graphs $G$ on $n$ vertices grows exponentially with $n$, by establishing that the (dual) flow polynomial…

A b-chromatic colouring of a graph $G$ is a proper $k$-colouring of the vertices of $G$, for some integer $k$, such that, for each colour $i$ ($1\leq i\leq k$), there exists a vertex $v$ of colour $i$ such that $v$ is adjacent to a vertex…

A hole in a graph $G$ is an induced cycle of length at least four, and a $k$-multihole in $G$ is a set of pairwise disjoint and nonadjacent holes. It is well known that if $G$ does not contain any holes then its chromatic number is equal to…

A Gallai coloring of the complete graph is an edge-coloring with no rainbow triangle. This concept first appeared in the study of comparability graphs and anti-Ramsey theory. We introduce a transitive analogue for acyclic directed graphs,…

DP-coloring (also called correspondence coloring) is a generalization of list coloring recently introduced by Dvo\v{r}\'{a}k and Postle. Several known bounds for the list chromatic number of a graph $G$, $\chi_\ell(G)$, also hold for the…

A gain graph is a graph whose edges are labelled invertibly by "gains" from a group. "Switching" is a transformation of gain graphs that generalizes conjugation in a group. A "weak chromatic function" of gain graphs with gains in a fixed…

The chromatic symmetric function $X_G$ is a sum of monomials corresponding to proper vertex colorings of a graph $G$. Crew, Pechenik, and Spirkl (2023) recently introduced a $K$-theoretic analogue $\overline{X}_G$ called the Kromatic…

DP-coloring (also called correspondence coloring) is a generalization of list coloring that has been widely studied in recent years after its introduction by Dvo\v{r}\'{a}k and Postle in 2015. As the analogue of the chromatic polynomial of…

A vertex colouring $f:V(G)\to C$ of a graph $G$ is complete if for any $c_1,c_2\in C$ with $c_1\ne c_2$ there are in $G$ adjacent vertices $v_1,v_2$ such that $f(v_1)=c_1$ and $f(v_2)=c_2$. The achromatic number of $G$ is the maximum number…

Let $G$ be a graph. An acyclic $k$-coloring of $G$ is a map $c:V(G)\rightarrow \{1,\dots,k\}$ such that $c(u)\neq c(v)$ for any $uv\in E(G)$ and the subgraph induced by the vertices of any two colors $i,j\in \{1,\dots,k\}$ is a forest. If…

A $b$-coloring of a graph is a proper coloring such that every color class contains a vertex adjacent to at least one vertex in each of the other color classes. The $b$-chromatic number of a graph $G$, denoted by $b(G)$, is the maximum…

A complete partition of a graph $G$ is a partition of the vertex set such that there is at least one edge between any two parts. The largest $r$ such that $G$ has a complete partition into $r$ parts, each of which is an independent set, is…

The chromatic polynomial is characterized as the unique polynomial invariant of graphs, compatible with two interacting bialgebras structures: the first coproduct is given by partitions of vertices into two parts, the second one by a…

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and…

Let $f: V(G)\cup E(G)\rightarrow \{1,2,\dots,k\}$ be a non-proper total $k$-coloring of $G$. Define a weight function on total coloring as $$\phi(x)=f(x)+\sum\limits_{e\ni x}f(e)+\sum\limits_{y\in N(x)}f(y),$$ where $N(x)=\{y\in V(G)|xy\in…

While solving a question on list coloring of planar graphs, Dvo\v{r}\'{a}k and Postle introduced the new notion of DP-coloring (they called it correspondence coloring). A DP-coloring of a graph $G$ reduces the problem of finding a coloring…