  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.
