A Linear-Time Algorithm for 4-Coloring Some Classes of Planar Graphs.

Zuosong Liang,Huandi Wei,Heng Liu

doi:10.1155/2021/7667656

Abstract

Every graph G=(V, E) considered in this paper consists of a finite set V of vertices and a finite set E of edges, together with an incidence function that associates each edge e ∈ E of G with an unordered pair of vertices of G which are called the ends of the edge e. 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. A proper k-vertex-coloring of a graph G=(V, E) is a mapping c : V⟶S (S is a set of k colors) such that no two adjacent vertices are assigned the same colors. The famous Four Color Theorem states that a planar graph has a proper vertex-coloring with four colors. However, the current known proof for the Four Color Theorem is computer assisted. In addition, the correctness of the proof is still lengthy and complicated. In 2010, a simple O(n2) time algorithm was provided to 4-color a 3-colorable planar graph. In this paper, we give an improved linear-time algorithm to either output a proper 4-coloring of G or conclude that G is not 3-colorable when an arbitrary planar graph G is given. Using this algorithm, we can get the proper 4-colorings of 3-colorable planar graphs, planar graphs with maximum degree at most five, and claw-free planar graphs.

Highlights

Adding edges joining every vertex of G to every vertex of Computational Intelligence and Neuroscience
H, one obtains the join of G and H, denoted by G ∨ H. e join Cn ∨ K1 of a cycle Cn and a single vertex is referred to as a n-wheel denoted by Wn
A proper k-vertex-coloring, or a proper k-coloring, of a graph G (V, E) is a mapping c: V ⟶ S (S is a set of k colors) such that no two adjacent vertices are assigned the same colors

Summary

Introduction

Adding edges joining every vertex of G to every vertex of Computational Intelligence and Neuroscience. Given a planar graph G, we design an improved linear-time algorithm to either output a 4coloring of G or conclude that G is not 3-colorable. For a simple plane graph G, we call a vertex v of degree five bad if all faces incident with v, except for the at most one, are triangles and the exceptional face has size at most five.

Results

Conclusion