The Total Coloring Conjecture holds for planar graphs without three special 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.
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.