Odd vertex magic total labeling of the extended comb graph

Abstract

Let $G$ be a simple finite graph with $n$ vertices and  $m$  edges. A vertex magic total labeling is a bijection  $f$ from $V(G)\cup E(G)$ to the integers $\{1, 2, 3, \ldots, m + n\}$ with the property that for every $v$ in $V(G)$, $f(v)+\Sigma f(uv)= k$ for some constant $k$, where the sum is taken over all edges incident with $v$. The parameter $k$ is called the  magic constant for $f$.  Nagaraj et al. (C. T. Nagaraj, C. Y. Ponnappan and  G. Prabakaran, Odd vertex magic total labeling of trees, International Journal of Mathematics Trends and Technology, 52(6), 2017, 374-379) introduced the concept of odd vertex magic total labeling. A vertex magic total labeling is called an  odd vertex magic total labeling  if  $f(V(G)) = \{1, 3, 5, \ldots, 2n- 1\}$. A graph $G$ is called an odd vertex magic if there exists an odd vertex magic total labeling for $G$. In this paper we prove that the extended comb graph $EC (t,k)  for  k=2$ admits an odd vertex magic total labeling when $t$  is odd and the extended comb graph $EC (t,k),  k=2$ with an additional edge admits an odd vertex magic total labeling when $t$  is even.