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

Abstract Let G be a graph with vertex set V ( G ) and edge set E ( G ) . A vertex labeling f : V ( G ) → Z 2 induces an edge labeling f + : E ( G ) → Z 2 defined by f + ( x y ) = f ( x ) + f ( y ) , for each edge x y ∈ E ( G ) . For i ∈ Z 2 , let v f ( i ) = | { v ∈ V ( G ) : f ( v ) = i } | and e f ( i ) = | { e ∈ E ( G ) : f + ( e ) = i } | . We say f is friendly if | v f ( 0 ) − v f ( 1 ) | ≤ 1 . We say G is cordial if | e f ( 1 ) − e f ( 0 ) | ≤ 1 for a friendly labeling f . The set F I ( G ) = { | e f ( 1 ) − e f ( 0 ) | : f  is friendly } is called the friendly index set of G . In this paper, we investigate the friendly index sets of the edge-gluing of a complete graph K n and n copies of cycles C 3 . The cordiality of the graphs is also determined.