Further results on product cordial graphs

A binary vertex labeling of graph $ G $ with induced edge labeling $ f^{*}:E(G) rightarrow {0,1} $ defined by $ f^{*}(e=uv)=f(u)f(v) $ is called a textit{product cordial labeling} if $ vert v_{f}(0) - v_{f}(1) vert leq 1 $ and $ vert e_{f}(0) - e_{f}(1) vert leq 1 $. A graph is called textit{product cordial} if it admits product cordial labeling. We prove that the shell admits a product cordial labeling. Sundaram et al.cite{Sundaram1} proved that if a graph with $p$ vertices and $ q$ edges with $ p geq 4 $ is product cordial then $ q leq dfrac{p^{2}-1}{4} + 1 $. We present here some families of graphs which satisfy this condition but not product cordial.