On k-totally magic cordial labeling of graphs

Let G = (V (G), E(G)) be a simple, finite and undirected graph of order n. Given a bijection f : V (G). E(G). Zk such that for each edge uv. E(G), f(u)+f(v)+f(uv) is constant C(mod k). Let nf (i) be the number of vertices and edges labeled by i under f. If | n(f) (i) -n(f) (j) <= 1 for all 0 = i <= j <= k-1, we say f is a k-totally magic cordial (k-TMC) labeling of G. A graph is said to be k-totally cordial magic if it admits a k-TMC labeling. In this paper, we give some ways to construct new families of k-TMC graphs from a known k-totally cordial magic graphs. We also give a sufficient condition for an odd graph to admit no k-TMC labeling. As a by-product, we determine the k-totally magic cordiality of many families of graphs.