Coloring the Cartesian sum of graphs

For graphs GG and HH, let G⊕HG⊕H denote their Cartesian sum. We investigate the chromatic number and the circular chromatic number for G⊕HG⊕H. It has been proved that for any graphs GG and HH, χ(G⊕H)≤max{⌈χc(G)χ(H)⌉,⌈χ(G)χc(H)⌉}. It has been conjectured that for any graphs GG and HH, χc(G⊕H)≤max{χ(H)χc(G),χ(G)χc(H)}. We confirm this conjecture for graphs GG and HH with special values of χc(G)χc(G) and χc(H)χc(H). These results improve previously known bounds on the corresponding coloring parameters for the Cartesian sum of graphs.