发表机构
Guizhou University; Yangzhou University（贵州大学; 扬州大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对路径和圈的广义Mycielski图，确定其游戏色数介于4至5之间，得出M_2(P_5)与M_2(P_6)的游戏色数精确值为4，证明其不随层数和基础图阶数增长而超过5。
AI 中文摘要
图着色游戏是一种双人游戏，两名玩家交替为图G中未着色的顶点着色；游戏色数是先手玩家保证获胜所需的最少颜色数。我们针对广义Mycielski图M_k(G)研究该参数，其中G是含n个顶点的路径P_n或圈C_n。对所有k≥2且n≥5，我们确定4≤χ_g(M_k(P_n))≤5，以及4≤χ_g(M_k(C_n))≤5。我们还得出精确值χ_g(M_2(P_5))=χ_g(M_2(P_6))=4。下界证明使用了Bob可同时制造两个威胁的构型，而两个精确情形下的四色上界则通过双引理（double-doctor lemma）证明。由此可见，尽管层数和基础图的阶数可能增长，但游戏色数仍被限制在5以内。
英文摘要
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 $k\geq2$ and $n\geq5$, we establish $4\leqχ_g\bigl(M_k(P_n)\bigr)\leq5$ and $4\leqχ_g\bigl(M_k(C_n)\bigr)\leq5$. We also determine the exact values $χ_g\bigl(M_2(P_5)\bigr)=χ_g\bigl(M_2(P_6)\bigr)=4$. 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.
Comments16 pages, 3 figures, 12 tables
