Related papers: Tree-colorable maximal planar graphs

An equitable tree-$k$-coloring of a graph is a vertex coloring on $k$ colors so that every color class incudes a forest and the sizes of any two color classes differ by at most one.This kind of coloring was first introduced in 2013 and can…

We show that any planar graph $G=(V,E)$ has a 5-coloring such that one color class contains at most $|V|/6$ vertices. In other words, there exists a partition of $V$ into five independent sets $\{V_1, \cdots, V_5\}$ such that $|V_5| \leq…

A coloring of a graph G = (V,E) is a partition {V1, V2, . . ., Vk} of V into independent sets or color classes. A vertex v Vi is a Grundy vertex if it is adjacent to at least one vertex in each color class Vj . A coloring is a Grundy…

An equitable $(t,k,d)$-tree-coloring of a graph $G$ is a coloring to vertices of $G$ such that the sizes of any two color classes differ by at most one and the subgraph induced by each color class is a forest of maximum degree at most $k$…

Given a graph $G$, a colouring of $G$ is \emph{acyclic} if it is a proper colouring of $G$ and every cycle contains at least three colours. Its acyclic chromatic number $\chi_a(G)$ is the minimum~$k$ such that an acyclic $k$-colouring of…

This paper proves the following result: Assume $G$ is a triangle free planar graph, $X$ is an independent set of $G$. If $L$ is a list assignment of $G$ such that $\mid L(v)\mid = 4$ for each vertex $v \in V(G)-X$ and $\mid L(v)\mid = 3$…

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…

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…

We prove the existence of a function $f :\mathbb{N} \to \mathbb{N}$ such that the vertices of every planar graph with maximum degree $\Delta$ can be 3-colored in such a way that each monochromatic component has at most $f(\Delta)$ vertices.…

DP-coloring of a simple graph is a generalization of list coloring, and also a generalization of signed coloring of signed graphs. It is known that for each $k \in \{3, 4, 5, 6\}$, every planar graph without $C_k$ is 4-choosable.…

For a graph G and an integer t we let mcc_t(G) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed…

Given a graph $G$, a vertex-colouring $\sigma$ of $G$, and a subset $X\subseteq V(G)$, a colour $x \in \sigma(X)$ is said to be \emph{odd} for $X$ in $\sigma$ if it has an odd number of occurrences in $X$. We say that $\sigma$ is an…

An odd coloring of a graph $G$ is a proper coloring such that any non-isolated vertex in $G$ has a coloring appears odd times on its neighbors. The odd chromatic number, denoted by $\chi_o(G)$, is the minimum number of colors that admits an…

A repetition is a sequence of symbols in which the first half is the same as the second half. An edge-coloring of a graph is repetition-free or nonrepetitive if there is no path with a color pattern that is a repetition. The minimum number…

This paper proves that for each positive integer $m$, there is a planar graph $G$ which is not $(4m+\lfloor \frac{2m-1}{9}\rfloor,m)$-choosable. Then we pose some conjectures concerning multiple list colouring of planar graphs.

For $1\leq s_1 \le s_2 \le \ldots \le s_k$ and a graph $G$, a packing $(s_1, s_2, \ldots, s_k)$-coloring of $G$ is a partition of $V(G)$ into sets $V_1, V_2, \ldots, V_k$ such that, for each $1\leq i \leq k$, the distance between any two…

A conjecture due to the fourth author states that every $d$-regular planar multigraph can be $d$-edge-coloured, provided that for every odd set $X$ of vertices, there are at least $d$ edges between $X$ and its complement. For $d = 3$ this…

Let $G$ be a planar graph without 4-cycles and 5-cycles and with maximum degree $\Delta\ge 32$. We prove that $\chi_{\ell}(G^2)\le \Delta+3$. For arbitrarily large maximum degree $\Delta$, there exist planar graphs $G_{\Delta}$ of girth 6…

DP-coloring (also known as correspondence coloring) of a simple graph is a generalization of list coloring. It is known that planar graphs without 4-cycles adjacent to triangles are 4-choosable, and planar graphs without 4-cycles are…

The \textit{square} of a graph $G$, denoted by $G^2$, is obtained from $G$ by adding an edge to connect every pair of vertices with a common neighbor in $G$. In this paper we prove that for every planar graph $G$ with maximum degree at most…