On $H$-irregularity strength of comb and edge comb product of graphs

Abstract

Let $G$ and $H$ be graphs and $1$ be a positive number. An $H$-irregular labeling of $G$ is an assignment of integers from $1$ up to $k$ to either vertices, edges, or both in $G$ such that each sum of labels in a subgraph isomorphic to $H$ are pairwise distinct. Moreover, a comb product of $G$ and $H$ is a construction of graph obtained by attaching several copies of $H$ to each vertices of $G$. Meanwhile, an edge comb product of $G$ and $H$ is an alternate construction where the copies of $H$ is attached on edges of $G$ instead. In this paper, we investigate the vertex, edge, and total $H$-irregular labeling of $G$ where both $G$ and $H$ is either a comb product or an edge comb product of graphs.