Related papers: Boundary chromatic polynomial
Given an integer base $b\geq 2$, a number $\rho\geq 1$ of colors, and a finite sequence $\Lambda=(\lambda_1,\ldots,\lambda_\rho)$ of positive integers, we introduce the concept of a $\Lambda$-restricted $\rho$-colored $b$-ary partition of…
We extend the Penrose polynomial, originally defined only for plane graphs, to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial of embedded graphs leads to new identities and relations for the Penrose polynomial…
This paper discusses reformulations of the problem of coloring plane maps with four colors. We give a number of alternate ways to formulate the coloring problem including a tautological expansion similar to the Penrose Bracket, and an…
The goal of this paper is to show the existence (using probabilistic tools) of configurations of lines, boxes, and points with certain interesting combinatorial properties. (i) First, we construct a family of $n$ lines in $\mathbb{R}^3$…
Chromatic quantum contextuality is a criterion of quantum nonclassicality based on (hyper)graph coloring constraints. If a quantum hypergraph requires more colors than the number of outcomes per maximal observable (context), it lacks a…
A vertex coloring $\phi$ of a graph $G$ is $p$-centered if for every connected subgraph $H$ of $G$ either $\phi$ uses more than $p$ colors on $H$ or there is a color that appears exactly once on $H$. Centered colorings form one of the…
For complete graphs and n-cubes bounds are found for the possible number of colours in an interval edge colourings.
In a recent paper by the same authors, we constructed a stationary 1-dependent 4-coloring of the integers that is invariant under permutations of the colors. This was the first stationary k-dependent q-coloring for any k and q. When the…
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…
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…
We show that determining the chromatic number of a $\{P_5,\bar{P_5}\}$-free graph or a $\{P_5,K_p-e\}$-free graph can be done in polynomial time
For a given number of colors, $s$, the guessing number of a graph is the (base $s$) logarithm of the cardinality of the largest family of colorings of the vertex set of the graph such that the color of each vertex can be determined from the…
As a fundamental metric for quantifying quantum advantage in non-local games, the quantum chromatic number reveals the power of entanglement in distributed tasks. In this paper, we investigate this parameter for $q$-ary Hamming graphs and a…
The beautiful Beraha-Kahane-Weiss theorem has found many applications within graph theory, allowing for the determination of the limits of root of graph polynomials in settings as vast as chromatic polynomials, network reliability, and…
We consider the problem of existence of perfect $2$-colorings (equitable $2$-partitions) of Hamming graphs with given parameters. We start with conditions on parameters of graphs and colorings that are necessary for their existence. Next we…
We investigate the linear chromatic number $\chi_{\text{lin}}(G(n,p))$ of the binomial random graph $G(n,p)$ on $n$ vertices in which each edge appears independently with probability $p=p(n)$. For dense random graphs ($np \to \infty$ as $n…
$(1^a, 2^b)$-coloring is the problem of partitioning the vertex set of a graph into $a$ independent sets and $b$ 2-independent sets. This problem was recently introduced by Choi and Liu. We study the computational complexity and extremal…
The graph coloring problem (GCP) is one of the most studied NP-HARD problems in computer science. Given a graph , the task is to assign a color to all vertices such that no vertices sharing an edge receive the same color and that the number…
A packing $k$-coloring for some integer $k$ of a graph $G=(V,E)$ is a mapping $\varphi:V\to\{1,\ldots,k\}$ such that any two vertices $u, v$ of color $\varphi(u)=\varphi(v)$ are in distance at least $\varphi(u)+1$. This concept is motivated…
We introduce a class of pairs of graphs consisting of two cliques joined by an arbitrary number of edges. The members of a pair have the property that the clique-bridging edge-set of one graph is the complement of that of the other. We…