On local antimagic total labeling of complete graphs amalgamation

IF 1 Q1 MATHEMATICS
G. Lau, W. Shiu
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引用次数: 0

Abstract

Let \(G = (V,E)\) be a connected simple graph of order \(p\) and size \(q\). A graph \(G\) is called local antimagic (total) if \(G\) admits a local antimagic (total) labeling. A bijection \(g : E \to \{1,2,\ldots,q\}\) is called a local antimagic labeling of $ if for any two adjacent vertices \(u\) and \(v\), we have \(g^+(u) \ne g^+(v)\), where \(g^+(u) = \sum_{e\in E(u)} g(e)\), and \(E(u)\) is the set of edges incident to \(u\). Similarly, a bijection \(f:V(G)\cup E(G)\to \{1,2,\ldots,p+q\}\) is called a local antimagic total labeling of \(G\) if for any two adjacent vertices \(u\) and \(v\), we have \(w_f(u)\ne w_f(v)\), where \(w_f(u) = f(u) + \sum_{e\in E(u)} f(e)\). Thus, any local antimagic (total) labeling induces a proper vertex coloring of \(G\) if vertex \(v\) is assigned the color \(g^+(v)\) (respectively, \(w_f(u)\)). The local antimagic (total) chromatic number, denoted \(\chi_{la}(G)\) (respectively \(\chi_{lat}(G)\)), is the minimum number of induced colors taken over local antimagic (total) labeling of \(G\). In this paper, we determined \(\chi_{lat}(G)\) where \(G\) is the amalgamation ofcomplete graphs. Consequently, we also obtained the local antimagic (total) chromatic number of the disjoint union of complete graphs, and the join of \(K_1\) and amalgamation of complete graphs under various conditions. An application of local antimagic total chromatic number is also given.
完全图合并的局部反幻全标记
设(G = (V,E))是一个阶为p,大小为q的连通简单图。如果图\(G\)允许一个局部反魔术(全)标记,则图\(G\)称为局部反魔术(全)。对于任意两个相邻的点\(u\)和\(v\),我们有\(g^+(u) \ne g^+(v)\),其中\(g^+(u) = \sum_{E \in E(u)} g(E)\),而\(E(u)\)是关联到\(u\)的边的集合。类似地,一个双射\(f:V(G)\杯E(G)\到\{1,2,\ldots,p+q\}\)被称为\(G\)的局部反奇异全标记,如果对于任意两个相邻的顶点\(u\)和\(V \),我们有\(w_f(u)\ne w_f(V)\),其中\(w_f(u) = f(u) + \sum_{E \in E(u)} f(E)\)。因此,如果顶点\(v\)被赋予颜色\(G ^+(v)\)(分别为\(w_f(u)\)),则任何局部反奇异(全)标记都会诱导出适当的顶点着色\(G\)。局部反魔术(总)色数表示为\(\chi_{la}(G)\)(分别为\(\chi_{lat}(G)\)),是占据\(G\)局部反魔术(总)标记的最小诱导色数。在本文中,我们确定了\(\chi_{lat}(G)\),其中\(G\)是完全图的合并。因此,我们也得到了完全图的不相交并的局部反奇异(全)色数,以及各种条件下完全图的K_1的连接和合并。给出了局部反幻全色数的一个应用。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Opuscula Mathematica
Opuscula Mathematica MATHEMATICS-
CiteScore
1.70
自引率
20.00%
发文量
30
审稿时长
22 weeks
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