Related papers: A structure of 1-planar graph and its applications…

We determine the maximum number of edges that a planar graph can have as a function of its maximum degree and matching number.

A graph $G$ is \emph{uniquely k-colorable} if the chromatic number of $G$ is $k$ and $G$ has only one $k$-coloring up to permutation of the colors. A uniquely $k$-colorable graph $G$ is edge-critical if $G-e$ is not a uniquely $k$-colorable…

We initiate the study of total-coloring extensions, and focus our attention on planar graphs, asking: ``When can a total-$k$-coloring of some subgraph $H$ of a planar graph $G$ be extended to a total-$k$-coloring of $G$?'' We prove that if…

Let $G$ be a simple planar graph of maximum degree $\Delta$, let $t$ be a positive integer, and let $L$ be an edge list assignment on $G$ with $|L(e)| \geq \Delta+t$ for all $e \in E(G)$. We prove that if $H$ is a subgraph of $G$ that has…

A drawing of a graph is $k$-plane if every edge contains at most $k$ crossings. A $k$-plane drawing is saturated if we cannot add any edge so that the drawing remains $k$-plane. It is well-known that saturated $0$-plane drawings, that is,…

For graphs of bounded maximum average degree, we consider the problem of 2-distance coloring. This is the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbor receive different…

We consider edge colorings of graphs. An edge coloring is a majority coloring if for every vertex at most half of the edges incident with it are in one color. And edge coloring is a distinguishing coloring if for every non-trivial…

This paper proves the following result: If $G$ is a planar graph and $L$ is a $4$-list assignment of $G$ such that $|L(x) \cap L(y)| \le 2$ for every edge $xy$, then $G$ is $L$-colourable. This answers a question asked by Kratochv\'{i}l,…

The inclusion relation between simple objects in the plane may be used to define geometric set systems, or hypergraphs. Properties of various types of colorings of these hypergraphs have been the subject of recent investigations, with…

$(1^a, 2^b)$-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…

Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology \cite{Civan}, and the framework through which it was studied, we introduce the linear coloring on graphs. We…

A graph is $k$-planar $(k \geq 1)$ if it can be drawn in the plane such that no edge is crossed more than $k$ times. A graph is $k$-quasi planar $(k \geq 2)$ if it can be drawn in the plane with no $k$ pairwise crossing edges. The families…

We define a perfect coloring of a graph $G$ as a proper coloring of $G$ such that every connected induced subgraph $H$ of $G$ uses exactly $\omega(H)$ many colors where $\omega(H)$ is the clique number of $H$. A graph is perfectly colorable…

A graph $G$ is said to be equitably $c$-colorable if its vertices can be partitioned into $c$ independent sets that pairwise differ in size by at most one. Chen, Lih, and Wu conjectured that every connected graph $G$ with maximum degree…

A \textit{$k$-total coloring} of a graph $G$ is an assignment of $k$ colors to its elements (vertices and edges) so that adjacent or incident elements have different colors. The total chromatic number is the smallest integer $k$ for which…

A graph $G$ is $(1,3)$-colorable if its vertices can be partitioned into subsets $V_1$ and $V_2$ so that every vertex in $G[V_1]$ has degree at most $1$ and every vertex in $G[V_2]$ has degree at most $3$. We prove that every graph with…

All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms…

A 1-planar graph refers to a graph that can be drawn on the plane such that each edge has at most one crossing. In this paper, focusing on the spectral Tur\'{a}n-type problems of $1$-planar graphs, we determine completely the unique…

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index $\chiup_{a}'(G)$ of a graph $G$ is the least number of colors in an acyclic edge…

Coloring a graph $G$ consists in finding an assignment of colors $c: V(G)\to\{1,\ldots,p\}$ such that any pair of adjacent vertices receives different colors. The minimum integer $p$ such that a coloring exists is called the chromatic…