| Summary: | The principal aim of this article is to initiate a study of the following coloring notion for digraphs. An odd <i>k</i>-edge coloring of a general digraph (directed pseudograph) <i>D</i> is a (not necessarily proper) coloring of its edges with at most <i>k</i> colors such that for every vertex <i>v</i> and color <i>c</i> holds: if <i>c</i> is used on the set <inline-formula><math display="inline"><semantics><mrow><msub><mi>∂</mi><mi>D</mi></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of edges incident with <i>v</i>, then <i>c</i> appears an odd number of times on each non-empty set from the pair <inline-formula><math display="inline"><semantics><mrow><msubsup><mi>∂</mi><mi>D</mi><mo>+</mo></msubsup><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>,</mo><msubsup><mi>∂</mi><mi>D</mi><mo>−</mo></msubsup><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of, respectively, outgoing and incoming edges incident with <i>v</i>. We show that it can be decided in polynomial time whether <i>D</i> admits an odd 2-edge coloring. Throughout the paper, several conjectures, questions and open problems are posed. In particular, we conjecture that for each odd edge-colorable digraph four colors suffice.
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