Coloring the Cartesian sum of graphs

For graphs G and H, let G ⊕ H denote their Cartesian sum. We investigate the chromatic number and the circular chromatic number for G ⊕ H. It is proved that for any graphs G and H, χ(G ⊕ H) ≤ max{dχc(G)χ(H)e, dχ(G)χc(H)e}. It is conjectured that for any graphs G and H, χc(G ⊕H) ≤ max{χ(H)χc(G), χ(G)χc(H)}. We confirm this conjecture for graphs G and H with special values of χc(G) and χc(H). These results improve previously known bounds on the corresponding coloring parameters for the Cartesian sum of graphs. 1991 Mathematics Subject Classification. 05C20, 05C35, 05C15 ∗Corresponding Author. Supported by the National Science Foundation, USA, under grant DMS 0302456. †Supported by the National Science Council, Taiwan, under grant NSC93-2115-M-110-004.