Abstract
The recursive maximal planar graphs can be obtained from K4, 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 (Δ+2)-colorable, where Δ 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 (Δ+1)-colorable. Moreover, we give linear time algorithms for total coloring of recursive maximal planar graphs and (2,2)-recursive maximal planar graphs, respectively.
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