The -independence number of direct products of graphs and Hedetniemi’s conjecture

The k -independence number of G , denoted as α k ( G ) , is the size of a largest k -colorable subgraph of G . The direct product of graphs G and H , denoted as G × H , is the graph with vertex set V ( G ) × V ( H ) , where two vertices ( x 1 , y 1 ) and ( x 2 , y 2 ) are adjacent in G × H , if x 1 is adjacent to x 2 in G and y 1 is adjacent to y 2 in H . We conjecture that for any graphs G and H , α k ( G × H ) ≤ α k ( G ) | V ( H ) | + α k ( H ) | V ( G ) | − α k ( G ) α k ( H ) . The conjecture is stronger than Hedetniemi’s conjecture. We prove the conjecture for k = 1 , 2 and prove that α k ( G × H ) ≤ α k ( G ) | V ( H ) | + α k ( H ) | V ( G ) | − α k ( G ) α ( H ) holds for any k .