New results on vertex equitable labeling

Abstract

The concept of vertex equitable labeling was introduced in [9]. A graph $G$ is said to be vertex equitable if there exists a vertex labeling $f$ such that for all $a$ and $b$ in $A$, $\left|v_f(a)-v_f(b)\right|\leq1$ and the induced edge labels are $1, 2, 3,\cdots, q$. A graph $G$ is said to be a vertex equitable if it admits a vertex equitable labeling. In this paper, we prove that the graphs, subdivision of double triangular snake $S(D(T_n))$, subdivision of double quadrilateral snake $S(D(Q_n))$, subdivision of double alternate triangular snake $S(DA(T_n))$, subdivision of double alternate quadrilateral snake $S(DA(Q_n))$, $DA(Q_m)\odot nK_1$ and $DA(T_m)\odot nK_1$ admit vertex equitable labeling.