On Graph Odd Edge-Colorings and Odd Edge-Coverings

An odd $k$-edge-coloring of a graph $G$ is a (not necessarily proper) edge-coloring with at most $k$ colors such that each non-empty color class induces a graph in which every vertex is of odd degree; similarly, if more than one color per edge is allowed, we speak of an odd $k$-edge-covering of $G$. In this paper, we fully resolve two major conjectures on odd edge-colorings and odd edge-coverings of graphs, proposed by Petru{š}evski and {Š}krekovski ({\it European Journal of Combinatorics,} 91:103225, 2021). The first conjecture states that, apart from two particular exceptions which are respectively odd $5$- and odd-$6$-edge-colorable, for any other loopless and connected graph $G$ there exists an edge $e$ such that $G\mathrm{\backslash }\{e\}$ is odd $3$-edge-colorable. The second conjecture states that any simple graph $G$ admits an odd $3$-edge-covering in which at most one edge receives more than one color. In addition, we strongly confirm the second conjecture by demonstrating that there exists an odd $3$-edge-covering in which at most one edge receives two colors and the rest of the edges receive unique colors.