Some Characterizations and NP-Complete Problems for Power Cordial Graphs

Abstract

A power cordial labeling of a graph $G=\left(V\left(G\right),E\left(G\right)\right)$ is a bijection $f:V\left(G\right)⟶\left\{\mathrm{1,2},\dots ,\left|V\left(G\right)\right|\right\}$ such that an edge $e=uv$ is assigned the label 1 if $f\left(u\right)={\left(f\left(v\right)\right)}^n$ or $f\left(v\right)={\left(f\left(u\right)\right)}^n$ , for some $n\in \mathbb{N}\cup \left\{0\right\}$ and the label 0 otherwise, and satisfy the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. The graph that admits power cordial labeling is called a power cordial graph. In this paper, we derive some characterizations of power cordial graphs as well as explore NP-complete problems for power cordial labeling. This work also rules out any possibility of forbidden subgraph characterization for power cordial labeling.