Inayah Nur, Susanto Faisal, Semaničová-Feňovčíková Andrea
Department of Mathematics, Faculty of Science and Technology, Syarif Hidayatullah State Islamic University Jakarta, Jalan Ir. H. Djuanda 95, Ciputat 15412, Indonesia.
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Tadulako, Jalan Soekarno Hatta Km. 9, Palu 94118, Indonesia.
Heliyon. 2021 Oct 18;7(10):e08203. doi: 10.1016/j.heliyon.2021.e08203. eCollection 2021 Oct.
An --antimagic total labeling of a simple graph admitting an -covering is a bijection such that for all subgraphs of isomorphic to , the set of -weights given by forms an arithmetic sequence where , are two fixed integers and is the number of all subgraphs of isomorphic to . Moreover, such a labeling is called super if the smallest possible labels appear on the vertices. A (super) --antimagic graph is a graph that admits a (super) --antimagic total labeling. In this paper the existence of super --antimagic total labelings for the -shadow and the closed -shadow of a connected for several values of is proved.
对于一个允许(r)-覆盖的简单图(G)，一个((k,d))-反魔法全标号是一个双射(f:V(G)\cup E(G)\to{1,2,\cdots,|V(G)| + |E(G)|})，使得对于(G)的所有同构于(rK_2)的子图(H)，由(w_f(H)=\sum_{x\in V(H)}f(x)+\sum_{e\in E(H)}f(e))给出的(r)-权重集合形成一个等差数列，其中(k,d)是两个固定整数，(n)是(G)中所有同构于(rK_2)的子图的数量。此外，如果最小可能的标签出现在顶点上，则这样的标号(f)称为超级的。一个（超级）((k,d))-反魔法图是一个允许（超级）((k,d))-反魔法全标号的图。在本文中，证明了对于几个(r)值，连通图(G)的(r)-影子和闭(r)-影子存在超级((k,d))-反魔法全标号。
