The game chromatic number of generalized Mycielski graphs of paths and cycles

The graph coloring game is a two-player game in which the players alternately color an uncolored vertex of a graph $G$. The game chromatic number is the minimum number of colors needed for the first player to guarantee a win. We investigate this parameter for generalized Mycielski graphs $M_k(G)$, where $G$ is a path $P_n$ or a cycle $C_n$ with $n$ vertices. For every $k\ge 2$ and $n\ge 5$, we establish $4\le {\chi }_g(M_k(P_n))\le 5$ and $4\le {\chi }_g(M_k(C_n))\le 5$. We also determine the exact values ${\chi }_g(M_2(P_5))={\chi }_g(M_2(P_6))=4$. The proofs of the lower bounds use a configuration in which Bob can create two threats simultaneously, while the four-color upper bounds in the two exact cases are proved using the double-doctor lemma. Thus the number of layers and the order of the base graph may grow, but the game chromatic number remains bounded by five.