Coloring the Cartesian Sum of Graphs

For graphs G and H, let G ⊕ H denote their Cartesian sum. We prove that for any graphs G and H, χ ( G ⊕ H ) ⩽ max { ⌈ χ c ( G ) χ ( H ) ⌉ , ⌈ χ ( G ) χ c ( H ) ⌉ } . Moreover the bound is sharp. We conjecture that for any graphs G and H, χ c ( G ⊕ H ) ⩽ max { χ ( H ) χ c ( G ) , χ ( G ) χ c ( H ) } . The conjectured bound would be sharp if it is true. We confirm this conjecture for graphs G and H with special values of χ c ( G ) and χ c ( H ) . These results improve previously known bounds on the chromatic number and the circular chromatic number for the Cartesian sum of graphs.