Strict Neighbor-Distinguishing Total Index of Graphs

A total-coloring of a graph G is strict neighbor-distinguishing if for any two adjacent vertices u and v, the set of colors used on u and its incident edges and the set of colors used on v and its incident edges are not included with each other. The strict neighbor-distinguishing total index χsnd′′(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{{\rm{snd}}}^{\prime \prime }(G)$$\end{document} of G is the minimum number of colors in a strict neighbor-distinguishing total-coloring of G. In this paper, we prove that every simple graph G with ∆(G) ≥ 3 satisfies χsnd′′(G)≤2Δ(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{{\rm{snd}}}^{\prime \prime }(G) \le 2\Delta (G)$$\end{document}.