相关论文: Counting Packings of List-colorings of Graphs

List colouring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list-colouring, we seek many in parallel. Our explorations have uncovered a…

The classic enumerative functions for counting colorings of a graph $G$, such as the chromatic polynomial $P(G,k)$, do so under the assumption that the given graph is labeled. In 1985, Hanlon defined and studied the chromatic polynomial for…

The chromatic polynomial of a graph $G$, denoted $P(G,m)$, is equal to the number of proper $m$-colorings of $G$. The list color function of graph $G$, denoted $P_{\ell}(G,m)$, is a list analogue of the chromatic polynomial that has been…

List packing is a notion that was introduced in 2021 (by Cambie et al.). The list packing number of a graph $G$, denoted $\chi_{\ell}^*(G)$, is the least $k$ such that for any list assignment $L$ that assigns $k$ colors to each vertex of…

The chromatic polynomial $\pi_{G}(k)$ of a graph $G$ can be viewed as counting the number of vertices in a family of coloring graphs $\mathcal C_k(G)$ associated with (proper) $k$-colorings of $G$ as a function of the number of colors $k$.…

The chromatic polynomial of a graph $G$, denoted $P(G,m)$, is equal to the number of proper $m$-colorings of $G$. The list color function of graph $G$, denoted $P_{\ell}(G,m)$, is a list analogue of the chromatic polynomial that has been…

For a graph $G$ and a list assignment $L$ with $|L(v)|=k$ for all $v$, an $L$-packing consists of $L$-colorings $\varphi_1,\cdots,\varphi_k$ such that $\varphi_i(v)\ne\varphi_j(v)$ for all $v$ and all distinct $i,j\in\{1,\ldots,k\}$. Let…

A colouring of a graph $G=(V,E)$ is a mapping $c\colon V\to \{1,2,\ldots\}$ such that $c(u)\neq c(v)$ for every two adjacent vertices $u$ and $v$ of $G$. The {\sc List $k$-Colouring} problem is to decide whether a graph $G=(V,E)$ with a…

One of Thomassen's classical results is that every planar graph of girth at least $5$ is 3-choosable. One can wonder if for a planar graph $G$ of girth sufficiently large and a $3$-list-assignment $L$, one can do even better. Can one find…

The chromatic polynomial of a graph is an important notion in algebraic combinatorics that was introduced by Birkhoff in 1912; denoted $P(G,k)$, it equals the number of proper $k$-colorings of graph $G$. Enumerative analogues of the…

DP-coloring (also called correspondence coloring) is a generalization of list coloring that was introduced by Dvo\v{r}\'{a}k and Postle in 2015. The chromatic polynomial of a graph is an important notion in algebraic combinatorics that was…

The chromatic polynomial of a graph $G$, denoted $P(G,m)$, is equal to the number of proper $m$-colorings of $G$ for each $m \in \mathbb{N}$. In 1990, Kostochka and Sidorenko introduced the list color function of graph $G$, denoted…

In [J. Combin. Theory Ser. B 161 (2023), 109--119], the authors showed that the list-color function $P_l(G,k)$ of any simple graph $G$ of size $m$ coincides with its chromatic polynomial $P(G,k)$ for all integers $k\ge m-1$. In this…

The fractional list packing number $\chi_{\ell}^{\bullet}(G)$ of a graph $G$ is a graph invariant that has recently arisen from the study of disjoint list-colourings. It measures how large the lists of a list-assignment $L:V(G)\rightarrow…

We study two weighted graph coloring problems, in which one assigns $q$ colors to the vertices of a graph such that adjacent vertices have different colors, with a vertex weighting $w$ that either disfavors or favors a given color. We…

In this paper, we study the problem of partitioning a graph into connected and colored components called blocks. Using bivariate generating functions and combinatorial techniques, we determine the expected number of blocks when the vertices…

The distinguishing number of a graph $G$, denoted $D(G)$, is the minimum number of colors needed to produce a coloring of the vertices of $G$ so that every nontrivial isomorphism interchanges vertices of different colors. A list assignment…

Proportional choosability is a list coloring analogue of equitable coloring. Specifically, a $k$-assignment $L$ for a graph $G$ specifies a list $L(v)$ of $k$ available colors to each $v \in V(G)$. An $L$-coloring assigns a color to each…

An equitable coloring is a proper coloring of a graph such that the sizes of the color classes differ by at most one. A graph $G$ is equitably $k$-colorable if there exists an equitable coloring of $G$ which uses $k$ colors, each one…

A colouring of a graph $G=(V,E)$ is a function $c: V\rightarrow\{1,2,\ldots \}$ such that $c(u)\neq c(v)$ for every $uv\in E$. A $k$-regular list assignment of $G$ is a function $L$ with domain $V$ such that for every $u\in V$, $L(u)$ is a…