Pair Difference Cordiality of Certain Subdivision Graphs

다국어 초록 (Multilingual Abstract)

\noindent Let $G = (V, E)$ be a $(p,q)$ graph.\\ Define \begin{equation*} \rho = \begin{cases} \frac{p}{2} ,& \text{if $p$ is even}\\ \frac{p-1}{2} ,& \text{if $p$ is odd}\\ \end{cases} \end{equation*}\\ and $L = \{\pm1 ,\pm2, \pm3 , \cdots ,\pm\rho\}$ called the set of labels.\\ \noindent Consider a mapping $f : V \longrightarrow L$ by assigning different labels in $L$ to the different elements of $V$ when $p$ is even and different labels in $L$ to $p-1$ elements of $V$ and repeating a label for the remaining one vertex when $p$ is odd.The labeling as defined above is said to be a pair difference cordial labeling if for each edge $uv$ of $G$ there exists a labeling $\left|f(u) - f(v)\right|$ such that $\left|\Delta_{f_1} - \Delta_{f_1^c}\right| \leq 1$, where $\Delta_{f_1}$ and $\Delta_{f_1^c}$ respectively denote the number of edges labeled with $1$ and number of edges not labeled with $1$. A graph $G$ for which there exists a pair difference cordial labeling is called a pair difference cordial graph. In this paper we investigate the pair difference cordial labeling behavior of subdivision of some graphs.