Many variations of the classical graph coloring model have been intensively studied due to their multiple applications; scheduling problems and aircraft assignments, for instance, motivate the robust coloring problem. This model gets to capture natural constraints of those optimization problems by combining the information provided by two colorings: a vertex coloring of a graph and the induced edge coloring on a subgraph of its complement; the goal is to minimize, among all proper colorings of the graph for a fixed number of colors, the number of edges in the subgraph with the endpoints of the same color. The study of the robust coloring model has been focused on the search for heuristics due to its NP-hard character when using at least three colors, but little progress has been made in other directions. We present a new approach on the problem obtaining the first collection of non-heuristic results for general graphs; among them, we prove that robust coloring is the model that better approaches the equitable partition of the vertex set, even when the graph does not admit a so-called \emph{equitable coloring}. We also show the NP-completeness of its decision problem for the unsolved case of two colors, obtain bounds on the associated robust coloring parameter, and solve a conjecture on paths that illustrates the complexity of studying this coloring model.

翻译：经典图着色模型的许多变体因其多重应用而受到深入研究；例如，调度问题和飞机分配问题催生了鲁棒着色问题。该模型通过结合两种着色提供的信息来捕捉这些优化问题的自然约束：图的顶点着色及其补图子图上的诱导边着色；其目标是在固定颜色数的所有正常着色中，最小化子图中端点颜色相同的边数。由于使用至少三种颜色时具有NP难特性，鲁棒着色模型的研究一直集中于启发式算法的探索，但在其他方向上进展甚微。我们对该问题提出了一种新方法，得到了关于一般图的首组非启发式结果；其中，我们证明了鲁棒着色是更接近顶点集均衡划分的模型，即使当图不存在所谓的均衡着色时也是如此。我们还证明了双色未解情况下的决策问题是NP完全的，获得了相关鲁棒着色参数的界，并解决了路径图上的一个猜想，这说明了研究该着色模型的复杂性。