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

The k-independence number of G, denoted as αk(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 (x1,y1) and (x2,y2) are adjacent in G×H, if x1 is adjacent to x2 in G and y1 is adjacent to y2 in H. We conjecture that for any graphs G and H, αk(G×H)≤αk(G)|V(H)|+αk(H)|V(G)|−αk(G)αk(H). The conjecture is stronger than Hedetniemi's conjecture. We prove the conjecture for k=1,2 and prove that αk(G×H)≤αk(G)|V(H)|+αk(H)|V(G)|−αk(G)α(H) holds for any k.