Group mean cordial labeling of some splitting graphs

Abstract

Let $G$ be a $(p, q)$ graph and let $A$ be a group. Let $f: V(G) \longrightarrow A$ be a map. For each edge $u v$ assign the label  $\left\lfloor\frac{o(f(u))+o(f(v))}{2} \mid\right.$. Here $o(f(u))$ denotes the order of $f(u)$ as an element of the group $A$. Let $\mathbb{I}$ be the set of all integers that are labels of the edges of $G$. $f$ is called a group mean cordial labeling if the following conditions hold:
(1) For $x, y \in A,\left|v_f(x)-v_f(y)\right| \leq 1$, where $v_f(x)$ is the number of vertices labeled with $x$.
(2) For $i, j \in \mathbb{I},\left|e_f(i)-e_f(j)\right| \leq 1$, where $e_f(i)$ denote the number of edges labeled with $i$.
A graph with a group mean cordial labeling is called a group mean cordial graph. In this paper, we take $A$ as the group of fourth roots of unity and prove that,the splitting graphs of Path $\left(P_n\right), \operatorname{Cycle}\left(C_n\right), \operatorname{Comb}\left(P_n \odot K_1\right)$ and Complete Bipartite graph ( $K_{n, n}$ when $n$ is even ) are group mean cordial graphs. Also we characterized the group mean cordial labeling of the splitting graph of $K_{1, n}$.