Related papers: Counting Colored Tilings on Grids and Graphs

In this paper, we study a graph parameter that was recently introduced, the burning number, focusing on a few probabilistic aspects of the problem. The original burning number is revisited and analyzed for binomial random graphs G(n,p),…

As the class $\mathcal T_4$ of graphs of twin-width at most 4 contains every finite subgraph of the infinite grid and every graph obtained by subdividing each edge of an $n$-vertex graph at least $2 \log n$ times, most NP-hard graph…

A proper coloring of vertices of a graph is equitable if the sizes of any two color classes differ by at most 1. Such colorings have many applications and are interesting by themselves. In this paper, we discuss the state of art and…

In this paper, we propose a new family of graphs, matrix graphs, whose vertex set $\mathbb{F}^{N\times n}_q$ is the set of all $N\times n$ matrices over a finite field $\mathbb{F}_q$ for any positive integers $N$ and $n$. And any two…

For any fixed surface Sigma of genus g, we give an algorithm to decide whether a graph G of girth at least five embedded in Sigma is colorable from an assignment of lists of size three in time O(|V(G)|). Furthermore, we can allow a subgraph…

We prove that the two-colouring number of any planar graph is at most 8. This resolves a question of Kierstead et al. [SIAM J. Discrete Math.~23 (2009), 1548--1560]. The result is optimal.

In this paper we consider Contact graphs of Paths on a Grid (CPG graphs), i.e. graphs for which there exists a family of interiorly disjoint paths on a grid in one-to-one correspondence with their vertex set such that two vertices are…

The Unfriendly Partition Problem asks whether it is possible to split the vertex set of an infinite graph $G$ into two parts so that every vertex has at least as many neighbors in the other part than on its own. Despite the uncountable…

We show that any complete $k$-partite graph $G$ on $n$ vertices, with $k \ge 3$, whose edges are two-coloured, can be covered with two vertex-disjoint monochromatic paths of distinct colours. We prove this under the necessary assumption…

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…

This work studies certain aspects of graphs embedded on surfaces. Initially, a colored graph model for a map of a graph on a surface is developed. Then, a concept analogous to (and extending) planar graph is introduced in the same spirit as…

The Unfriendly Partition Conjecture posits that every countable graph admits a 2-colouring in which for each vertex there are at least as many bichromatic edges containing that vertex as monochromatic ones. This is not known in general, but…

For a positive integer $k$ and graph $G=(V,E)$, a $k$-colouring of $G$ is a mapping $c: V\rightarrow\{1,2,\ldots,k\}$ such that $c(u)\neq c(v)$ whenever $uv\in E$. The $k$-Colouring problem is to decide, for a given $G$, whether a…

Let $P$ be a set of $n\geq 4$ points in general position in the plane. Consider all the closed straight line segments with both endpoints in $P$. Suppose that these segments are colored with the rule that disjoint segments receive different…

A vertex coloring of a graph $G$ is called distinguishing (or symmetry breaking) if no non-identity automorphism of $G$ preserves it, and the distinguishing number, shown by $D(G)$, is the smallest number of colors required for such a…

For a fixed integer, the $k$-Colouring problem is to decide if the vertices of a graph can be coloured with at most $k$ colours for an integer $k$, such that no two adjacent vertices are coloured alike. A graph $G$ is $H$-free if $G$ does…

An edge coloring of the $n$-vertex complete graph, $K_n$, is a Gallai coloring if it does not contain any rainbow triangle, that is, a triangle whose edges are colored with three distinct colors. We prove that for $n$ large and every $k$…

Given a graph $G=(V, E)$ and a list of available colors $L(v)$ for each vertex $v\in V$, where $L(v) \subseteq \{1, 2, \ldots, k\}$, List $k$-Coloring refers to the problem of assigning colors to the vertices of $G$ so that each vertex…

A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. A connected graph $G$ is said to be $t$-admissible if admits a special spanning tree in which the distance between any two adjacent vertices…

A b-coloring of a graph $G$ is a coloring of its vertices such that every color class contains a vertex that has neighbors in all other classes. The b-chromatic number of $G$ is the largest integer $k$ such that $G$ has a b-coloring with…