Set-defined graph classes: $\chi$-boundedness meets tropical algebra

Abstract

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 $\chi$-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 $\chi$-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 $\chi$-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 $\chi$-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-$\chi$-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.