Product-cordial index and friendly index of regular graphs

Let G=(V,E) be a connected simple graph‎. ‎A labeling f‎: ‎V\to Z_۲ induces two edge labelings f^+‎, ‎f^*‎: ‎E \to‎ ‎Z_۲ defined by f^+(xy) = f(x)+f(y) and f^*(xy) =‎ ‎f(x)f(y) for each xy \in E‎. ‎For i \in Z_۲‎, ‎let‎ ‎v_f(i) = |f^{-۱}(i)|‎, ‎e_{f^+}(i) = |(f^{+})^{-۱}(i)|‎ ‎and e_{f^*}(i) = |(f^*)^{-۱}(i)|‎. ‎A labeling f is‎ ‎called friendly if |v_f(۱)-v_f(۰)| \le ۱‎. ‎For a friendly‎ ‎labeling f of a graph G‎, ‎the friendly index of G‎ ‎under f is defined by i^+_f(G) = e_{f^+}(۱)-e_{f^+}(۰)‎. ‎The set \{i^+_f(G)\;|\;f \mbox{ 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^*}(۱)-e_{f^*}(۰)‎. ‎The set‎ ‎\{i^*_f(G)\;|\;f \mbox{ 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 Kwong‎, ‎Lee‎ ‎and Ng in ۲۰۱۰; and those of cycles‎.

Keywords:

friendly labeling , friendly index set , product-cordial index , product-cordial index set , Torus