| Abstract: | An odd k‐edge‐coloring of a graph G is a (not necessarily proper) edge‐coloring with at most k colors such that each nonempty color class induces a graph in which every vertex is of odd degree. Pyber (1991) showed that every simple graph is odd 4‐edge‐colorable, and Lužar et al. (2015) showed that connected loopless graphs are odd 5‐edge‐colorable, with one particular exception that is odd 6‐edge‐colorable. In this article, we prove that connected loopless graphs are odd 4‐edge‐colorable, with two particular exceptions that are respectively odd 5‐ and odd 6‐edge‐colorable. Moreover, a color class can be reduced to a size at most 2. |
