Circular chromatic number of Cartesian product of signed graphs

This paper studies the circular coloring of signed graphs. A signed graph is a graph with a signature that assigns a sign to each edge, either positive or negative. This paper studies circular coloring and a circular chromatic number of Type 1 and Type Cartesian products. We shall prove the following results: The circular chromatic number of Cartesian product Type 1 $(G,\sigma )\mathrm{\square }(H,\tau )$ is ${\chi }_c(G\mathrm{\square }H,\sigma \mathrm{\square }\tau )=\max \{{\chi }_c(G,\sigma ),{\chi }_c(H,\tau )\}$ and the circular chromatic number of Cartesian product Type 2 $(G,\sigma ){\mathrm{\square }}^{\mathrm{\prime }}(H,\tau )$ satisfies ${\chi }_c(G{\mathrm{\square }}^{\mathrm{\prime }}H,\sigma {\mathrm{\square }}^{\mathrm{\prime }}\tau )\le 2\max \{{\chi }_c(G),{\chi }_c(H)\}$.