AI 中文总结
研究集合定义的图类的χ-有界性，证明分解定理，给出全集定义类χ-有界性的二分法及判定算法，该算法将问题简化为热带线性规划可行性，建立了图论与热带代数、博弈论算法的联系。
AI 中文摘要
我们研究集合定义的图类，即顶点被赋予固定长度数值元组且邻接性仅由坐标间等式模式决定的遗传类。这类图出现在结构图论、通信复杂性、逻辑和邻接标签方案中。我们探讨其何时为χ-有界的，即整个类的色数是否由团数界定。首先证明了一个分解定理，每个集合定义类中的图可分解为团数的多项式有界部分，每个部分诱导出有限个移位可着色图的并集，移位可着色图是可同态到移位图的图。有界的移位可着色图并集是集合定义类中χ-有界性的基本障碍。对于由固定布尔规则在等式模式上实现的所有图组成的全集定义类，证明了更强的二分法：每个此类要么是多项式χ-有界的，要么包含任意大色数的移位图。此外，给出了一个算法，给定全集定义类的布尔函数描述，可判定该类的χ-有界性。它将问题简化为热带线性规划的可行性，其正确性源于与平均收益博弈中获胜策略的对偶性。反之，每个热带不等式的整数系统以及每个平均收益博弈都可在强多项式时间内编码为一个集合定义类，其非χ-有界性等同于可行性。这提供了热带可行性和平均收益博弈可解性的图论对应，连接了结构图论、热带代数和博弈论算法。
英文摘要
We study set-defined graph classes: hereditary classes whose vertices are assigned fixed-length numerical tuples, with adjacency determined solely by equality patterns among coordinates. These classes arise in structural graph theory, communication complexity, logic, and adjacency labeling schemes. We ask when they are $χ$-bounded, that is, when chromatic number is bounded in terms of clique number throughout the class. First, we prove a decomposition theorem: every graph in a set-defined class can be partitioned into a number of parts polynomially bounded in its clique number, each inducing a union of a bounded number of shift-colorable graphs, that is, graphs admitting a homomorphism to a shift graph. Thus bounded unions of shift-colorable graphs form the fundamental obstruction to $χ$-boundedness in set-defined classes. For full set-defined classes, consisting of all graphs realizable by a fixed Boolean rule on equality patterns, we prove a stronger dichotomy: every such class is either polynomially $χ$-bounded or contains shift graphs of arbitrarily large chromatic number. Moreover, we provide an algorithm that, given a Boolean-function description of a full set-defined class, decides $χ$-boundedness of the class. It reduces the problem to feasibility of tropical linear programs, and its correctness follows from a duality with winning strategies in mean-payoff games. Conversely, every integer system of tropical inequalities, and hence every mean-payoff game, can be encoded in strongly polynomial time as a set-defined class whose non-$χ$-boundedness is equivalent to feasibility. This provides a graph-theoretic counterpart of tropical feasibility and mean-payoff-game solvability, linking structural graph theory, tropical algebra, and game-theoretic algorithms.
