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In this paper, we fully resolve two major conjectures on odd edge-colorings and odd edge-coverings of graphs, proposed by Petrusevski and Skrekovski (European Journal of Combinatorics, 91: 103225, 2021). The first conjecture states that for any loopless and connected graph G with ₎₃₃' (G) =4, there exists an edge e such that G \e\ is odd 3-edge-colorable. The second conjecture states that any simple graph G with ₎₃₃' (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.
Liu et al. (Fri,) studied this question.