Menu
Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \). Let \( A = \{0, 1\} \). A labeling \( f: V(G) \to A \) induces a partial edge labeling \( f^*: E(G) \to A \) defined by
\[
f^*(xy) = f(x) \quad \text{if and only if } f(x) = f(y),
\]
for each edge \( xy \in E(G) \). For \( i \in A \), let
\[
v_f(i) = \text{card}\{v \in V(G) : f(v) = i\}
\]
and
\[
e_{f^*}(i) = \text{card}\{e \in E(G) : f^*(e) = i\}.
\]
A labeling \( f \) of a graph \( G \) is said to be friendly if
\[
\lvert v_f(0) – v_f(1) \rvert \leq 1.
\]
If
\[
\lvert e_{f^*}(0) – e_{f^*}(1) \rvert \leq 1,
\]
then \( G \) is said to be \textbf{balanced}. The balancedness of the Cartesian product and composition of graphs is studied in [19]. We provide some new families of balanced graphs using other constructions.