A linear-time algorithm for clique-coloring planar graphs

Abstract

A clique of a graph $G$ is a set of pairwise adjacent vertices of $G$. A clique-coloring of $G$ is an assignment of colors to the vertices of $G$ such that no inclusion-wise maximal clique of size at least 2 is monochromatic. Mohar and Škrekovski proved that planar graphs are 3-clique-colorable (Electr. J. Combin. 6 (1999), #R26). In this paper we present a linear time algorithm for 3-clique-coloring planar graphs by giving a new proof that planar graphs are 3-clique-colorable.

Introduction

All graphs considered here are finite, simple and nonempty. Let $G=(V,E)$ be a graph with vertex set $V$ and edge set $E$. For a vertex $v\in V$, the open neighborhood $N\left(v\right)$ of $v$ is defined as the set of vertices adjacent to $v$, i.e., $N\left(v\right)=\{u\mid uv\in E\}$. The closed neighborhood of $v$ is $N\left[v\right]=N\left(v\right)\cup \left\{v\right\}$. The degree of $v$ is equal to $\left|N\left(v\right)\right|$, denoted by $d_G\left(v\right)$ or simply $d\left(v\right)$. By $\delta \left(G\right)$ and $\Delta \left(G\right)$, we denote the minimum degree and the maximum degree of the graph $G$ respectively. For a subset $S\subseteq V$, the subgraph induced by $S$ is denoted by $G\left[S\right]$.

A clique of a graph $G$ is a set of pairwise adjacent vertices of $G$. A clique on $m$ vertices is called an $m$-clique of $G$. A clique-coloring of $G$, also called weak coloring in the literature, is an assignment of colors to the vertices of $G$ in such a way that no inclusion-wise maximal clique of size at least two of $G$ is monochromatic. A $k$-clique-coloring of $G$ is a clique-coloring $\phi$: $V\to \{1,2,\dots ,k\}$ of $G$. If $G$ has a $k$-clique-coloring, we say that $G$ is $k$-clique-colorable. The clique-chromatic number of $G$, denoted by ${\chi }_C\left(G\right)$, is the smallest integer $k$ such that $G$ is $k$-clique-colorable. Clearly, every proper vertex coloring of $G$ is also a clique-coloring, and so ${\chi }_C\left(G\right)\le \chi \left(G\right)$. Furthermore, if a graph $G$ is triangle-free, then the clique-coloring of $G$ coincides with its proper vertex coloring. This implies that there are triangle-free graphs of arbitrarily large clique-chromatic number since there are triangle-free graphs of arbitrarily large chromatic number. In general, clique-coloring can be a very different problem from ordinary vertex coloring. The most notable difference is that clique-coloring has not a hereditary property: it is possible that a graph is $k$-clique-colorable, but it has an induced subgraph that is not [2]. Another difference is that a large clique is not an obstruction for clique colorability: even 2-clique-colorable graphs can contain arbitrarily large cliques. Determining ${\chi }_C\left(G\right)$ is hard: to decide ${\chi }_C=2$ is NP-complete on 3-chromatic perfect graphs [16], graphs with maximum degree $3$ [2] and even ($K_4$, diamond)-free perfect graphs [10]. Clique-coloring has received considerable attention (see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]).

A graph is said to be a planar graph, if it can be drawn in the plane so that its edges intersect only at their ends. Such a drawing is called a planar embedding of the graph. Any such particular embedding is called a plane graph.

Planar graphs were proved to be 3-clique-colorable, i.e., ${\chi }_C\le 3$ [21] and ${\chi }_C=2$ can be tested in polynomial time if the input is restricted to planar instances [16]. In [24] it was proved that every claw-free planar graph, different from an odd cycle, is 2-clique-colorable and a polynomial-time algorithm for finding the clique-coloring was proposed. In this note we provide a linear time algorithm for $3$-clique-coloring an arbitrary planar graph by giving a new proof that planar graphs are $3$-clique-colorable.