The edge-irregularity deletion conjecture

Let $G=(V,E)$ be a connected graph, and let $m=|E|$ be its number of edges. An edge-irregulator of $G$ is a set $S\subseteq E$ such that $G-S$ is locally irregular, meaning that every edge of $G-S$ joins vertices of different degrees; write ${\rm I}_e(G)$ for the minimum possible size of an edge-irregulator. Edge-irregularity deletion conjecture. There is an absolute constant $c\geq 1$ such that, for every connected graph $G$ with $m$ edges,

$${\rm I}_e(G)\leq \frac{1}{3}m+c.$$

The conjecture is motivated by the behaviour of cycles, where deleting one edge can resolve at most three local degree conflicts, while small graphs such as $K_2$ and $C_4$ show that an additive constant is necessary. The general bound remains open in the source paper.