The Total Coloring Conjecture holds for planar graphs without three special subgraphs

TL;DR

This paper proves that the Total Coloring Conjecture is valid for planar graphs with maximum degree six that do not contain three specific subgraphs, extending previous results to a broader class of graphs.

Contribution

It establishes the validity of TCC for a new class of planar graphs excluding three particular substructures, improving upon prior partial results.

Findings

TCC holds for planar graphs without mushroom, tent, and cone subgraphs.

The result applies to graphs with maximum degree six.

It includes graphs with certain sparse fan and wheel subgraphs.

Abstract

The Total Coloring Conjecture (TCC) for planar graphs with a maximum degree of six remains open. Previous studies suggest that TCC is valid for such graphs if they do not contain any subgraph isomorphic to a 4-fan. In this paper, we present an improved conclusion by establishing that TCC holds for planar graphs that are free of three particular substructures, namely the mushroom, the tent, and the cone. This advancement enhances previous findings by demonstrating that TCC is applicable to planar graphs with a maximum degree of six, which can accommodate sparse 4-fans, 5-fans, 5-wheels, and 6-wheels.