  In this paper, we fully resolve two major conjectures on odd edge-colorings
and odd edge-coverings of graphs, proposed by Petru{\v{s}}evski and
{\v{S}}krekovski ({\it European Journal of Combinatorics,} 91:103225, 2021).
The first conjecture states that for any loopless and connected graph $G$ with
$\chi_{\text{odd}}'(G)=4$, there exists an edge $e$ such that $G\backslash
\{e\}$ is odd $3$-edge-colorable. The second conjecture states that any simple
graph $G$ with $\chi_{\text{odd}}'(G)=4$ admits an odd $3$-edge-covering in
which only one edge receives two or three colors. In addition, we strongly
confirm the second conjecture by demonstrating that there exists an odd
$3$-edge-covering in which only one edge receives two colors.
