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

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${v_n^k}$$\end{document} represents the number of vertices of degree n which lie on k distinct 3-cycles, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n, k \in \mathbb{N}}$$\end{document} , then the conjecture is true for planar graphs which satisfy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${v_5^4 +2(v_5^{5^+} +v_6^4) +3v_6^5 +4v_6^{6^+} < 24}$$\end{document}.