Total vertex product irregularity strength of graphs

Abstract

Consider a simple graph $G$. We call a labeling $w:E(G)\cup V(G)\to \{1,2,\dots ,s\}$ (\textit{total vertex}) \textit{product-irregular}, if all product degrees $p{d}_G(v)$ induced by this labeling are distinct, where $p{d}_G(v)=w(v)\times {\prod }_{e\ni v}w(e)$. The strength of $w$ is $s$, the maximum number used to label the members of $E(G)\cup V(G)$. The minimum value of $s$ that allows some irregular labeling is called \textit{the total vertex product irregularity strength} and denoted $tvps(G)$. We provide some general bounds, as well as exact values for chosen families of graphs. Keywords: product-irregular labeling, total vertex product irregularity strength, vertex-distinguishing labeling.