Murugan Varadhan

A graph $G$ is antimagic if there exists a bijection $f$ from $E(G)$ to $\left\{1,2, \dots,|E(G)|\right\}$ such that the vertex sums for all vertices of $G$ are distinct, where the vertex sum is defined as the sum of the labels of all incident edges. Hartsfield and Ringel conjectured that every connected graph other than $K_2$ admits an antimagic labeling. It is still a challenging problem to address antimagicness in the case of disconnected graphs. In this paper, we study antimagicness for the disconnected graph that is constructed as the direct product of a star and a path.

An antimagic labeling of a graph $G=(V,E)$ is a bijection from the set of edges of G to $\left\{1,2,\cdots ,\left|E\left(G\right)\right|\right\}$ and such that any two vertices of G have distinct vertex sums where the vertex sum of a vertex v in $V\left(G\right)$ is nothing but the sum of all the incident edge labeling of G. In this paper, we discussed the antimagicness of rooted product and corona product of graphs. We proved that if we let G be a connected t-regular graph and H be a connected k-regular graph, then the rooted product of graph G and H admits antimagic labeling if $t\ge k$. Moreover, we proved that if we let G be a connected t-regular graph and H be a connected k-regular graph, then the corona product of graph G and H admits antimagic labeling for all $t,k\ge 2$.