Faudree–Lehel conjecture on the irregularity strength of regular graphs

Let $G$ be a $d$-regular graph with $n$ vertices, where $d\geq 2$. Its irregularity strength $s(G)$ is the least positive integer $k$ for which there is a weighting $\omega:E(G)\to\{1,2,\ldots,k\}$ such that the weighted degrees

$$\sigma_\omega(v)=\sum_{u\in N_G(v)}\omega(uv)$$

are distinct for all vertices $v$ of $G$. Faudree–Lehel conjecture. There exists an absolute constant $C$ such that

$$s(G)\leq\frac{n}{d}+C.$$

The conjecture asks whether the lower bound of order $n/d$ for regular graphs is asymptotically sharp up to an additive constant. The source describes this as a longstanding open problem, while known general bounds give a constant-factor estimate.