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) ∪ E(G) to the integers {1, 2, 3, . . . , m + n} with the property that for every v in V (G), f(v) + Σf(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, . . . , 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.