Generalized Odd-Even Sum Labeling and Some $\alpha-$Odd-Even Sum Graphs

Abstract

A $(p, q)$ graph $G$ is said to be an $\alpha$-odd-even sum graph if it admits an odd-even sum labeling $f$ defined by Monika and Murugan [9] by adding an addition condition that there is a positive integer $k(0<k<2 q-1)$ such that for every edge $u v \in E(G), \min (f(u), f(v))<k<\max \left(f(u), f(v)\right.$ ). In this paper, we study $\alpha$-odd-even sum labeling of $C_n(n \equiv 0$ (mod 4)), $S\left(x_1, x_2, \ldots, x_n\right), K_{m, n}(m, n \geq 2), P_n \square P_m(m, n \geq 2)$, step grid graph $S t_n(n \geq 3)$ and splitting graph of $K_{1, n}$.