Labeling Subgraph Embeddings and Cordiality of Graphs

Let G be a graph with vertex set V (G) and edge set E(G), a vertex labeling f : V (G) ! Z2 induces an edge labeling f+ : E(G) → Z2 defined by f+(xy) = f(x) + f(y), for each edge xy ∈ E(G). For each i ∈ Z2, let vf (i) = |{u ∈ V (G) : f(u) = i}| and ef+(i) = |{xy ∈ E(G) : f+(xy) = i}|. A vertex labeling f of a graph G is said to be friendly if |vf (1) − vf (0)| 1. The friendly index set of the graph G, denoted by FI(G), is defined as {|ef+(1) − ef+(0)| : the vertex labeling f is friendly}. The full friendly index set of the graph G, denoted by FFI(G), is defined as {ef+(1) − ef+(0) : the vertex labeling f is friendly}. A graph G is cordial if −1, 0 or 1 ∈ FFI(G). In this paper, by introducing labeling subgraph embeddings method, we determine the cordiality of a family of cubic graphs which are double-edge blow-up of P2 × Pn, n ≥ 2. Consequently, we completely determined friendly index and full product cordial index sets of this family of graphs.