On Graph Odd Edge-Colorings and Odd Edge-Coverings

In this paper, we fully resolve two major conjectures on odd edge-colorings and odd edge-coverings of graphs, proposed by Petru{{s}}evski and {{S}}krekovski ({ European Journal of Combinatorics,} 91:103225, 2021). The first conjecture states that for any loopless and connected graph G with χodd'(G)=4, there exists an edge e such that G \ is odd $3$-edge-colorable. The second conjecture states that any simple graph G with χodd'(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.