Antimagic Labeling of the Lexicographic Product Graph Km,n[Pk]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{m,n}[P_k]$$\end{document}

A labeling f of a graph G is a bijection from its edge set E(G) to the set {1,2,…,|E(G)|}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{1, 2, \ldots , |E(G)|\}$$\end{document}, which is antimagic if the vertex-sums are pairwise distinct, where the vertex-sum at one vertex is the sum of labels of all edges incident to such vertex. A graph is called antimagic if it admits an antimagic labeling f. In this paper, we show that the graph Km,n[Pk]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{m,n}[P_{k}]$$\end{document}, which is the lexicographic product of the complete bipartite graph Km,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{m,n}$$\end{document} and path Pk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{k}$$\end{document}, is antimagic.