Permutation Labeling of Graphs

Abstract

A $(p,q)$-graph is said to be a permutation graph if there exists a bijection function $f:V(G)\to \{1,2,\dots ,p\}$ such that the induced edge function $h_f:E(G)\to \mathbb{N}$ is defined as follows:
$$h_f(x_i,x_j)=\{\begin{array}{ll}{}^{f(x_i)}{P}_{f(x_j)},& \text{if }f(x_j)<f(x_i);\\ {}^{f(x_j)}{P}_{f(x_i)},& \text{if }f(x_i)<f(x_j).\end{array}$$ In this paper, we investigate the permutation labelings of wheel-related graphs.