On friendly index sets of the edge-gluing of complete graph and cycles

Let GG be a graph with vertex set V(G)V(G) and edge set E(G)E(G). A vertex labeling f:V(G)→Z2f:V(G)→Z2 induces an edge labeling f+:E(G)→Z2f+:E(G)→Z2 defined by f+(xy)=f(x)+f(y)f+(xy)=f(x)+f(y), for each edge xy∈E(G)xy∈E(G). For i∈Z2i∈Z2, let vf(i)=|{v∈V(G):f(v)=i}|vf(i)=|{v∈V(G):f(v)=i}| and ef(i)=|{e∈E(G):f+(e)=i}|ef(i)=|{e∈E(G):f+(e)=i}|. We say ff is friendly if |vf(0)−vf(1)|≤1|vf(0)−vf(1)|≤1. We say GG is cordial if |ef(1)−ef(0)|≤1|ef(1)−ef(0)|≤1 for a friendly labeling ff. The set FI(G)={|ef(1)−ef(0)|:f  is friendly}FI(G)={|ef(1)−ef(0)|:f  is friendly} is called the friendly index set of GG. In this paper, we investigate the friendly index sets of the edge-gluing of a complete graph KnKn and nn copies of cycles C3C3. The cordiality of the graphs is also determined.