A sufficient condition for planar graphs with maximum degree eight to be totally 9-colorable
Abstract
A total coloring of a graph G is a coloring of the vertices and edges such that two adjacent or incident elements receive different colors. The minimum number of colors required for a total coloring of a graph G is called the total chromatic number, denoted by ''(G). Let G be a planar graph of maximum degree eight. It is known that 9≤ ''(G) ≤ 10. We here prove that ''(G)=9 when the graph does not contain any subgraph isomorphic to a 4-fan.
0
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.