A signed graph is a pair of a graph and a mapping from the edge set to \(\{+1,-1\}\) . In 1982, Zaslavsky introduced the notion of a proper coloring of signed graphs as a natural generalization of a proper coloring of unsigned graphs. An odd coloring of a graph is a proper coloring of a graph such that every non-isolated vertex has a color that appears at an odd number of neighbors. This notion was introduced by Petruševski and Škrekovski in 2022, and has been actively studied. As a common generalization of these two concepts, in this paper, we introduce the notion of odd coloring of signed graphs. As an analogy of the Heawood’s map-color problem, for signed graphs embedded in a closed surface, we show that (1) for every closed surface S, every signed graph embedded in S is odd 2H(S)-colorable, and that (2) for every closed surface other than the Klein bottle, there is a signed graph with the odd chromatic number \(2H(S)-1\) that can be embedded in S, where H(S) denotes the Heawood number of S.
