The k-independence number of direct products of graphs and Hedetniemi's conjectureŠpacapan, Simon (Avtor) matematikateorija grafovneodvisnostno številokartezični produkt grafovmathematicsgraph theoryindependence numberCartesian product of graphsThe ▫$k$▫-independence number of ▫$G$▫, denoted as ▫$alpha_k(G)$▫, is the size of a largest ▫$k$▫-colorable subgraph of ▫$G$▫. The direct product of graphs ▫$G$▫ and ▫$H$▫, denoted as ▫$G times H$▫, is the graph with vertex set ▫$V(G) times V(H)$▫, where two vertices ▫$(x_1, y_1)$▫ and ▫$(x_2, y_2)$▫ are adjacent in ▫$G times H$▫, 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 ▫$H$▫, ▫$$alpha_k(G times H) ge alpha_k(G)|V(H)| + alpha_k(H)|V(G)| - alpha_k(G) alpha_k(H).$$▫ The conjecture is stronger than Hedetniemi's conjecture. We prove the conjecture for ▫$k = 1, 2$▫ and prove that ▫$alpha_k(G times H) ge alpha_k(G)|V(H)| + alpha_k(H)|V(G)| - alpha_k(G) alpha_k(H)$▫ holds for any ▫$k$▫.20112015-07-10 15:23:39Delo ni kategorizirano51910sl