Odd mean labeling of the graphs $P_{a,b}, P_a^b$ and $P_{\left\langle 2a\right\rangle}^b$

R. Vasuki   ; A. Nagarajan

Kragujevac Journal of Mathematics, Tome 36 (2012) no. 1, p. 141

Résumé

Let $G(V,E)$ be a graph with $p$ vertices and $q$ edges. A graph $G$ is said to be odd mean if there exists a function $f:V(G)\rightarrow \{0,1,2,3,\dots,2q-1\}$ satisfying $f$ is $1-1$ and the induced map $f^*:E(G)\rightarrow\{1,3,5,\dots,2q-1\}$ defined by \begin{equation*} f^*(uv)=eft\{\begin{array}{ll} \frac{f(u)+f(v)}{2}\quad\mbox{if $f(u)+f(v)$ is even} [2mm] \frac{f(u)+f(v)+1}{2}\quad\mbox{if $f(u)+f(v)$ is odd}\end{array}\right. \end{equation*} is a bijection. If a graph $G$ admits an odd mean labeling then $G$ is called an odd mean graph. In this paper we study the odd meanness of the class of graphs $P_{a,b}, P_a^b$ and $P_{\left\langle 2a\right\rangle}^b$ and we prove that the graphs $P_{2r,m}, P_{2r+1, 2m+1}, P_{2r}^m, P_{2r+1}^{2m+1}$ and $P_{\left\langle 2r,m\right\rangle}$ for all values of $r$ and $m$ are odd mean graphs.

Détail

Classification : 05C78
Keywords: Labeling, Odd mean labeling, Odd mean graphs