Parity and strong parity edge-colorings of graphs

A parity walk in an edge-coloring of a graph is a walk along which each color is used an even number of times. A parity edge-coloring (respectively, strong parity edge-coloring) is an edge-coloring in which there is no nontrivial parity path (respectively, open parity walk). The parity edge-chromatic number p(G) (respectively, strong parity edge-chromatic number $widehat{p}(G)$ ) is the least number of colors in a parity edge-coloring (respectively, strong parity edge-coloring) of G. Notice that $widehat{p}(G) ge p(G) ge chi'(G) ge Delta(G)$ for any graph G. In this paper, we determine $widehat{p}(G)$ and p(G) for some complete bipartite graphs and some products of graphs. For instance, we determine $widehat{p}(K_{m,n})$ and p(K _m,n ) for m≤n with n≡0,?1,?2 (mod 2^?lg?m?).