Vertex equitable labeling of cycle and path related graphs (Q2811825)

For a graph \(G\) and the set \(A=\left\{0,1,2,\dots,\left\lceil\frac {|E(G)|}2\right\rceil\right\}\), a vertex labelling \(f:V(G)\rightarrow A\) \textit{induces} an edge labelling \(f^*:E(G)\rightarrow\{1,2,\dots\}\) defined by \(f^*(uv)=f(u)+f(v)\) for all \(uv\in E(G)\). \(G\) is \textit{vertex equitable} if there exists a labelling \(f\) (said to be \textit{``vertex equitable''}) such that \(\left|f^{-1}(u)-f^{-1}(v)\right|\leq1\) for all \(u,v\in A\), and if \(f^*(E(G))=\{1,2,\dots,|E(G)|\}\). Extending results of \textit{M. Seenivasan} and \textit{A. Lourdusamy} [J. Discrete Math. Sci. Cryptography 11, No. 6, 727--735 (2008; Zbl 1178.05084)], the authors find vertex equitable labellings of graphs related to cycles and paths.