A 2-edge-colored graph is a pair \((G, \sigma )\) where G is a graph, and \(\sigma :E(G)\rightarrow \{\text {'}+\text {'},\text {'}-\text {'}\}\) is a function which marks all edges with signs. A 2-edge-colored coloring of the 2-edge-colored graph \((G, \sigma )\) is a homomorphism into a 2-edge-colored graph \((H, \delta )\). The 2-edge-colored chromatic number of the 2-edge-colored graph \((G, \sigma )\) is the minimum order of H. In this paper we show that for every 2-dimensional grid \((G, \sigma )\) there exists a homomorphism from \((G, \sigma )\) into the 2-edge-colored Paley graph \(SP_9\). Hence, the 2-edge-colored chromatic number of the 2-edge-colored grids is at most 9. This improves the upper bound on this number obtained recently by Bensmail. Additionally, we show that 2-edge-colored chromatic number of the 2-edge-colored grids with 3 columns is at most 8.

2边色图是一对\（（G，\ sigma）\），其中G是图，而\（\ sigma：E（G）\ rightarrow \ {\ text {'} + \ text {' }，\ text {'}-\ text {'} \} \）是用符号标记所有边缘的函数。2边色图\（（G，\ sigma）\）的2边色着色是2边色图\（（H，\ delta）\）的同态。2边彩色图\（（G，\ sigma）\）的2边彩色色数是H的最小阶。在本文中，我们表明，对于每个二维网格\（（G，\ sigma）\），存在从\（（（G，\ sigma）\）到2边色Paley图的同态\（SP_9 \）。因此，2边彩色网格的2边彩色色数最多为9。这改善了Bensmail最近获得的该数字的上限。此外，我们显示具有3列的2边缘色网格的2边缘色色数最多为8。