Conjecture 4.4: Refuted in C. Tardif, I. Zaguia, Hedetniemi's conjecture and fiber products of graphs, AKCE Int. J. Graphs Comb. 9 (2012), 53-58.

Conjecture 6.1: Proved in X. Zhu, The fractional version of Hedetniemi's conjecture is true, European J. Combin. 32 (2011), 1168-1175.

Problem 6.4: Answered positively in C. Tardif, Hedetniemi's conjecture and dense boolean lattices, Order 28 (2011), 181-191.

Problem 7.1: Answered positively in M Wrochna, Square-free graphs are multiplicative, J. Combin. Theory, Ser. B 122 (2017), 479-507.

Problem 7.2: Answered positively in X. Zhu, The fractional version of Hedetniemi's conjecture is true, European J. Combin. 32 (2011), 1168-1175.