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A proper [k]-total coloring c of a graph G is a proper total coloring c of G using colors of the set [k]={1,2,…,k}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$[k]=\{1,2,\ldots ,k\}$$\end{document}. Let Σ(u)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Sigma (u)$$\end{document} denote the sum of the color on a vertex u and the colors on all the edges incident with u. For each edge uv∈E(G)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$uv\in E(G)$$\end{document}, if Σ(u)≠Σ(v)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Sigma (u)\ne \Sigma (v)$$\end{document}, then we say the coloring distinguishes adjacent vertices by sum and call it a neighbor sum distinguishing total coloring of G. By tndiΣ(G)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$tndi_{\Sigma }(G)$$\end{document}, we denote the minimal value of k in such a coloring of G. It has been conjectured by Pilśniak et al. that Δ(G)+3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Delta (G)+3$$\end{document} colors enable the existence of a neighbor sum distinguishing total coloring. In this paper, we consider the neighbor sum distinguishing total coloring of corona product G∘H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$G\circ H$$\end{document} and obtain that tndiΣ(G∘H)≤Δ(G∘H)+3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$tndi_{\Sigma }(G\circ H)\le \Delta (G\circ H)+3$$\end{document}.
Bulletin of the Malaysian Mathematical Sciences Society – Springer Journals
Published: Oct 27, 2020
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