Abstract
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 has been proved that for any graphs G and H , χ ( G ⊕ H ) ≤ max { ⌈ χ c ( G ) χ ( H ) ⌉ , ⌈ χ ( G ) χ c ( H ) ⌉ } . It has been conjectured that for any graphs G and H , χ c ( G ⊕ H ) ≤ max { χ ( H ) χ c ( G ) , χ ( G ) χ c ( H ) } . 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 corresponding coloring parameters for the Cartesian sum of graphs.
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