Finding irregular subgraphs via local adjustments

For a graph $H$, let $m(H,k)$ denote the number of vertices of degree $k$ in $H$. A conjecture of Alon and Wei states that for any $d\geq 3$, every $n$-vertex $d$-regular graph contains a spanning subgraph $H$ satisfying $|m(H,k)-\frac{n}{d+1}|\leq 2$ for every $0\leq k \leq d$. This holds easily when $d\leq 2$. An asymptotic version of this conjecture was initially established by Frieze, Gould, Karo\'nski and Pfender, subsequently improved by Alon and Wei, and most recently enhanced by Fox, Luo and Pham, approaching its complete range. All of these approaches relied on probabilistic methods. In this paper, we provide a novel framework to study this conjecture, based on localized deterministic techniques which we call local adjustments. We prove two main results. Firstly, we show that every $n$-vertex $d$-regular graph contains a spanning subgraph $H$ satisfying $|m(H,k)-\frac{n}{d+1}|\leq 2d^2$ for all $0\leq k \leq d$, which provides the first bound independent of the value of $n$. Secondly, we confirm the case $d=3$ of the Alon-Wei Conjecture in a strong form. Both results can be generalized to multigraphs and yield efficient algorithms for finding the desired subgraphs $H$. Furthermore, we explore a generalization of the Alon-Wei Conjecture for multigraphs and its connection to the Faudree-Lehel Conjecture concerning irregularity strength.