Some Results on Odd Mean Graphs

Abstract

Let $G=(V,E)$ be a graph with $p$ vertices and $q$ edges. A graph $G$ is said to have an odd mean labeling if there exists a function $f:V(G)\rightarrow\{0,1,2,\dots,2q-1\}$ satisfying $f$ is $1$-$1$ and the induced map $f^*:E(G)\rightarrow\{1,3,5,\dots,2q-1\}$ defined by
\[f^*(uv)=\left\{\begin{array}{ll}\frac{f(u)+f(v)}{2}&\quad\mbox{if $f(u)+f(v)$ is even}\\ \frac{f(u)+f(v)+1}{2}&\quad\mbox{if $f(u)+f(v)$ is odd.}\end{array}\right.\]
is a bijection. A graph that admits an odd mean labeling is called an odd mean graph. In this paper, we prove that the graphs slanting ladder $SL_n$ for $n\geq 2,$ $Q_n\odot K_1$ for $n\geq 1,$ $TW(P_{2n})$ for $n\geq 2, H_n\odot mK_1$ for all $n\geq 1, m\geq 1$ and $mQ_3$ for $m\geq 1$ are odd mean graphs.