Circular chromatic number of Cartesian product of signed graphs

Abstract: 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 $(G,\sigma)\Box (H,\tau)$ is $\chi_{c}(G \Box H,\sigma\Box\tau)=\max{\chi_{c}(G,\sigma),\chi_{c}(H,\tau)}$ and the circular chromatic number of Cartesian product Type 2 $(G,\sigma)\Box' (H,\tau)$ satisfies $\chi_{c}(G \Box' H,\sigma\Box' \tau)\leq 2\max{\chi_{c}(G),\chi_{c}(H)}$.