Total Vertex Irregularity Strength of Cubic and 4-Regular Graphs

Abstract

Let $G$ be a graph and $k$ be a positive integer. A total $k$-labeling of $G$ assigns to each vertex and each edge a label from $\{1,\ldots,k\}$. The weight of a vertex is the sum of its label and the labels of its incident edges. A total labeling is vertex irregular if all vertex weights are distinct. The total vertex irregularity strength $\text{tvs}(G)$ is the smallest $k$ for which $G$ has a vertex irregular total $k$-labeling. For an $r$-regular graph $G$ on $n$ vertices, a counting argument gives $\text{tvs}(G)\ge\lceil(n+r)/(r+1)\rceil$. The restriction of a conjecture of Nurdin, Baskoro, Salman, and Gaos to regular graphs asserts that this bound is attained. We prove this assertion for cubic and $4$-regular graphs. We also show that, for every fixed $r\ge2$, a recent theorem on prescribed degree frequencies implies the assertion for all sufficiently large $r$-regular graphs.