Related papers: List $4$-colouring of planar graphs
A graph is outer-1-planar if it can be drawn in the plane so that all vertices are on the outer face and each edge is crossed at most once. It is known that the list edge chromatic number $\chi'_l(G)$ of any outer-1-planar graph $G$ with…
The chromatic number of an planar graph is not greater than four and this is known by the famous four color theorem and is equal to two when the planar graph is bipartite. When the planar graph is even-triangulated or all cycles are greater…
A proper $k$-coloring of $G$ is called an odd coloring of $G$ if for every vertex $v$, there is a color that appears at an odd number of neighbors of $v$. This concept was introduced recently by Petru\v{s}evski and \v{S}krekovski, and they…
A graph G is dually chordal if there is a spanning tree T of G such that any maximal clique of G induces a subtree in T. This paper investigates the Colourability problem on dually chordal graphs. It will show that it is NP-complete in case…
A simpler proof of the four color theorem is presented. The proof was reached using a series of equivalent theorems. First the maximum number of edges of a planar graph is obatined as well as the minimum number of edges for a complete…
In 2012, L\'ev\^eque, Maffray, and Trotignon conjectured that each graph $G$ that contains no induced subdivision of $K_4$ is $4$-colorable. In this paper, we prove that this conjecture holds when $G$ contains a $K_{1,2,3}$.
It is known that DP-coloring is a generalization of a list coloring in simple graphs and many results in list coloring can be generalized in those of DP-coloring. In this work, we introduce a relaxed DP-coloring which is a generalization if…
A graph $G$ is said to be $k$-critical if $G$ is $k$-colorable and $G-e$ is not $k$-colorable for every edge $e$ of $G$. In this paper, we present some new methods from two or more small 4-critical graphs to construct a larger 4-critical…
For a number $l\geq 2$, let ${\cal{G}}_l$ denote the family of graphs which have girth $2l+1$ and have no odd hole with length greater than $2l+1$. Wu, Xu and Xu conjectured that every graph in $\bigcup_{l\geq 2} {\cal{G}}_{l}$ is…
DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced by Dvo\u{r}\'{a}k and Postle (2017). Recently, Huang et al. [https://doi.org/10.1016/j.amc.2019.124562] showed that planar graphs with…
A graph is $k$-planar if it can be drawn in the plane so that each edge is crossed at most $k$ times. Typically, the class of 1-planar graphs is among the most investigated graph families within the so-called "beyond planar graphs". A…
A graph $G$ is called uniquely k-list colorable (U$k$LC) if there exists a list of colors on its vertices, say $L=\lbrace S_v \mid v \in V(G) \rbrace $, each of size $k$, such that there is a unique proper list coloring of $G$ from this…
Assume $G$ is a graph. We view $G$ as a symmetric digraph, in which each edge $uv$ of $G$ is replaced by a pair of opposite arcs $e=(u,v)$ and $e^{-1}=(v,u)$. Assume $S$ is an inverse closed subset of permutations of positive integers. We…
List coloring generalizes graph coloring by requiring the color of a vertex to be selected from a list of colors specific to that vertex. One refinement of list coloring, called choosability with separation, requires that the intersection…
Assume $k$ is a positive integer, $\lambda=\{k_1, k_2, \ldots, k_q\}$ is a partition of $k$ and $G$ is a graph. A $\lambda$-list assignment of $G$ is a $k$-list assignment $L$ of $G$ such that the colour set $\cup_{v\in V(G)}L(v)$ can be…
DP-coloring (also known as correspondence coloring) is a generalization of list coloring, introduced by Dvo\v{r}\'ak and Postle in 2017. It is well-known that there are non-4-choosable planar graphs. Much attention has recently been put on…
In this paper we have given a unified graph coloring algorithm for planar graphs. The problems that have been considered in this context respectively, are vertex, edge, total and entire colorings of the planar graphs. The main tool in the…
In 1994, Thomassen famously proved that every planar graph is 5-choosable, resolving a conjecture initially posed by Vizing and, independently, Erd\H{os}, Rubin, and Taylor in the 1970s. Later, Thomassen proved that every planar graph of…
A graph is $(c_1, c_2, ..., c_k)$-colorable if the vertex set can be partitioned into $k$ sets $V_1,V_2, ..., V_k$, such that for every $i: 1\leq i\leq k$ the subgraph $G[V_i]$ has maximum degree at most $c_i$. We show that every planar…
A graph is $\ell$-choosable if, for any choice of lists of $\ell$ colors for each vertex, there is a list coloring, which is a coloring where each vertex receives a color from its list. We study complexity issues of choosability of graphs…