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 \(K_{1, n}\).