Edge $k$-Product Cordial Labeling of Graphs

Abstract

In this paper, we introduce a new labeling namely `edge $k$-product cordial labeling' as follows: For a graph $G=(V(G), E(G))$ having no isolated vertex, an edge labeling $f:E(G)\rightarrow \left\lbrace 0,1,...,k-1 \right\rbrace$, where $k>1$ is an integer, is said to be an edge k-product cordial labeling if it induces a vertex labeling $f^\star:V(G)\rightarrow \left\lbrace 0,1,...,k-1 \right\rbrace$ defined by $f^\star(v)=\prod_{uv\in E(G)}   f(uv)(mod ~k)$ satisfies $ \left| e_{f}(i)-e_{f}(j) \right|  \leq 1$ and $\left|v_{f^\star}(i)-v_{f^\star}(j) \right| \leq 1$ for $i,j \in \left\lbrace 0,1,..., k-1 \right\rbrace$, where $e_{f}(i)$ and $v_{f^\star}(i)$ denote the number of edges and vertices respectively having a label i ($i=0,1,...,k-1$). Further, we study the edge k-product cordial behavior of star, bistar, shadow and splitting graph of star, path union of star, bistar and cycle graphs.