Total coloring of recursive maximal planar graphs

The recursive maximal planar graphs can be obtained from K-4, by embedding a 3-vertex in a triangular face continuously. A total k-coloring of a graph G is a coloring of its vertices and edges such that no two adjacent or incident elements receive the same color. The Total Coloring Conjecture, in short, TCC, states that every simple graph G is totally (Delta + 2)-colorable, where Delta is the maximum degree of G. In this paper, we prove that TCC holds for recursive maximal planar graphs, especially, a main class of recursive maximal planar graphs, named (2,2)-recursive maximal planar graphs, are totally (Delta + 1)-colorable. Moreover, we give linear time algorithms for total coloring of recursive maximal planar graphs and (2,2)-recursive maximal planar graphs, respectively. (C) 2022 Elsevier B.V. All rights reserved.

Automated summary

This paper proves that the Total Coloring Conjecture holds for recursive maximal planar graphs, especially for a main class of recursive maximal planar graphs called (2,2)-recursive maximal planar graphs. Linear time algorithms for total coloring of recursive maximal planar graphs and (2,2)-recursive maximal planar graphs are also provided.