Cordial Labellings of the Cartesian Product and Composition of Graphs

Let \(G\) be a graph. A labelling \(f: V(G) \to \{0,1\}\) is called a binary labelling of \(G\). A binary labelling \(f\) of \(G\) induces an edge labelling \(\lambda\) of \(G\) as follows:
\[\lambda(u,v) = |f(u) – f(v)|\] \quad for every edge \(uv \in E(G)\).
Let \(v_f(0)\) and \(v_f(1)\) be the number of vertices of \(G\) labelled with \(0\) and \(1\) under \(f\), and \(e_0(0)\) and \(e_1(1)\) be the number of edges labelled with \(0\) and \(1\) under \(\lambda\), respectively. Then the binary labelling \(f\) of \(G\) is said to be cordial if
\[|v_f(0) – v_f(1)| \leq 1 \quad {and} \quad |e_f(0) – e_f(1)| \leq 1.\]

A graph \(G\) is cordial if it admits a cordial labelling.

In this paper, we shall give a sufficient condition for the Cartesian product \(G \times H\) of two graphs \(G\) and \(H\) to be cordial. The Cartesian product of two cordial graphs of even sizes is then shown to be cordial. We show that the Cartesian products \(P_n \times P_n\) for all \(n \geq 2\) and \(P_n \times C_{4m}\) for all \(m\) and all odd \(n\) are cordial. The Cartesian product of two even trees of equal order such that one of them has a \(2\)-tail is shown to be cordial. We shall also prove that the composition \(C_n[K_2]\) for \(n \geq 4\) is cordial if and only if \(n \not = 2 \pmod{4}\). The cordiality of compositions involving trees, unicyclic graphs, and some other graphs are also investigated.