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Product-cordial index and friendly index of regular graphs. (English) Zbl 1293.05342
Summary: Let \(G=(V,E)\) be a connected simple graph. A labeling \(f:V\to Z_2\) induces two edge labelings \(f^+, f^*: E \to Z_2\) defined by \(f^+(xy) = f(x)+f(y)\) and \(f^*(xy) = f(x)f(y)\) for each \(xy \in E\). For \(i \in Z_2\), let \(v_f(i) = |f^{-1}(i)|\), \(e_{f^+}(i) = |(f^{+})^{-1}(i)|\) and \(e_{f^*}(i) = |(f^*)^{-1}(i)|\). A labeling \(f\) is called friendly if \(|v_f(1)-v_f(0)| \leq 1\). For a friendly labeling \(f\) of a graph \(G\), the friendly index of \(G\) under \(f\) is defined by \(i^+_f(G) = e_{f^+}(1)-e_{f^+}(0)\). The set \(\{i^+_f(G) \mid f\mathrm{ is a friendly labeling of }G\}\) is called the full friendly index set of \(G\). Also, the product-cordial index of \(G\) under \(f\) is defined by \(i^*_f(G) = e_{f^*}(1)-e_{f^*}(0)\). The set \(\{i^*_f(G) \mid f\mathrm{ is a friendly labeling of }G\}\) is called the full product-cordial index set of \(G\). In this paper, we find a relation between the friendly index and the product-cordial index of a regular graph. As applications, we will determine the full product-cordial index sets of torus graphs which was asked by H. Kwong et al. [Congr. Numerantium 206, 139–150 (2010; Zbl 1218.05169)] and those of cycles.

MSC:
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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