Totally vertex-magic cordial labeling

Abstract

In this paper, we introduce a new labeling called Totally Vertex-Magic Cordial(TVMC) labeling. A graph $G(p,q)$ is said to be TVMC with a constant $C$ if there is a mapping $f:V(G)\cup E(G)\to \{0,1\}$ such that

$$\left[f(a)+{\displaystyle \sum _{b\in N(a)}f(ab)}\right]\equiv C(\mathrm{mod}2)$$

for all vertices $a\in V(G)$ and $|n_f(0)-n_f(1)|\le 1$, where $N(a)$ is the set of vertices adjacent to the vertex $a$ and $n_f(i)(i=0,1)$ is the sum of the number of vertices and edges with label $i$.