All Graphs Have Antimagic Total Labelings

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• 作者：Mirka  ; Miller ;   ; ^{mirka.miller@newcastle.edu.au} ; Oudone  ; Phanalasy ;   ; ^{oudone.phanalasy@uon.edu.au} ; Joe  ; Ryan ;   ; ^{joe.ryan@newcastle.edu.au}
• 关键词：Vertex Weight ; Antimagic Total Labeling ; Super Antimagic Total Labeling ; Repus Antimagic Total Labeling
• 刊名：Electronic Notes in Discrete Mathematics
• 出版年：2011
• 出版时间：1 December 2011
• 年：2011
• 卷：38
• 期：Complete
• 页码：645-650
• 全文大小：178 K

文摘

Let be a finite, simple, indirected and either connected or disconnected. A antimagic total labeling of a graph with p vertices and q edges is a bijection such that all vertex weights are pairwise distinct, where a vertex weight is the sum of the label of that vertex and labels of all edges incident with the vertex. A graph is antimagic total if it has an antimagic total labeling. An antimagic total labeling l of a graph is called super (resp. repus) if (resp. ). A graph is super (resp. repus) antimagic total if it has a super (resp. repus) antimagic total labeling. In this paper we prove that all graphs have antimagic total labelings. We also prove that all graphs have super antimagic total labelings and repus antimagic total labelings. Furthermore, we show that some graphs have super -antimagic total labelings and repus -antimagic total labelings, that is, labelings that use the p smallest labels (resp. p largest labels) on vertices and, moreover, the vertex weights form arithmetic progression.