A New Approach in Finding Full Friendly Indices

Abstract

Let G be a graph with vertex set V(G) and edge set E(G), a vertex labeling $$f : V(G)\rightarrow \mathbb {Z}_2$$ induces an edge labeling $$ f^{+} : E(G)\rightarrow \mathbb {Z}_2$$ defined by $$f^{+}(xy) = f(x) + f(y)$$ , for each edge $$ xy\in E(G)$$ . For each $$i \in \mathbb {Z}_2$$ , let $$ v_{f}(i)=|\{u \in V(G) : f(u) = i\}|$$ and $$e_{f^+}(i)=|\{xy\in E(G) : f^{+}(xy) = i\}|$$ . A vertex labeling f of a graph G is said to be friendly if $$| v_{f}(1)-v_{f}(0) | \le 1$$ . The friendly index set of the graph G, denoted by FI(G), is defined as $$\{|e_{f^+}(1) - e_{f^+}(0)|$$ : the vertex labeling f is friendly $$\}$$ . The full friendly index set of the graph G, denoted by FFI(G), is defined as $$\{e_{f^+}(1) - e_{f^+}(0)$$ : the vertex labeling f is friendly $$\}$$ . In this paper, we determine FFI(G) for a class of cubic graphs with full vertices blow-up of cycle by a complete tripartite graph K(1, 1, 2) using a new method known as embedding labeling graph method. As a by-product, we also discuss the cordiality and the full product-cordial index sets for this graph.