A Larger Family of Planar Graphs that Satisfy the Total Coloring Conjecture

Abstract

The article shrinks the Δ = 6 hole that exists in the family of planar graphs which satisfy the total coloring conjecture. Let G be a planar graph. If \({v_n^k}\) represents the number of vertices of degree n which lie on k distinct 3-cycles, for \({n, k \in \mathbb{N}}\) , then the conjecture is true for planar graphs which satisfy \({v_5^4 +2(v_5^{5^+} +v_6^4) +3v_6^5 +4v_6^{6^+} < 24}\).