On Balancedness of Some Graph Constructions

Let G be a graph with vertex set V(G) and edge set E(G), and let A = {O, I}. A labeling f: V(G) ~ A induces a partial edge labeling f* : E(G) ~ A defined by f*(xy) = f(x), if and only if f(x) = fey), for each edge xy E E(G). For i E A, let vrt:i) = card{v E V(G): ftv) = i} and ep(i) = card{e E E(G): f*(e) i}. A labeling f ofa graph G is said to be friendly if I vrt:O) vrt: I) 1 s: I. If 1 ep(O) ­ ef*( 1) 1 s: 1 then G is said to be balanced. Balancedness of the Cartesian product and composition ofgraphs is studied in [19]. We provide some new families of balanced graphs using other constructions.