This work investigates structural and computational aspects of list-based graph coloring under interval constraints. Building on the framework of analogous and p-analogous problems, we show that classical List Coloring, $μ$-coloring, and $(γ,μ)$-coloring share strong complexity-preserving correspondences on graph classes closed under pendant-vertex extensions. These equivalences allow hardness and tractability results to transfer directly among the models. Motivated by applications in scheduling and resource allocation with bounded ranges, we introduce the interval-restricted $k$-$(γ,μ)$-coloring model, where each vertex receives an interval of exactly $k$ consecutive admissible colors. We prove that, although $(γ,μ)$-coloring is NP-complete even on several well-structured graph classes, its $k$-restricted version becomes polynomial-time solvable for any fixed $k$. Extending this formulation, we define $k$-$(γ,μ)$-choosability and analyze its expressive power and computational limits. Our results show that the number of admissible list assignments is drastically reduced under interval constraints, yielding a more tractable alternative to classical choosability, even though the general decision problem remains located at high levels of the polynomial hierarchy. Overall, the paper provides a unified view of list-coloring variants through structural reductions, establishes new complexity bounds for interval-based models, and highlights the algorithmic advantages of imposing fixed-size consecutive color ranges.

翻译：本研究探讨了在区间约束下列表式图着色的结构与计算特性。基于类比问题与p-类比问题的框架，我们证明了经典的列表着色、$μ$-着色与$(γ,μ)$-着色在悬挂顶点扩展封闭的图类上具有保持复杂度的强对应关系。这些等价性使得困难性与可解性结果能够在不同模型间直接迁移。受限于范围的调度与资源分配应用启发，我们引入了区间限制的$k$-$(γ,μ)$-着色模型，其中每个顶点被分配恰好$k$种连续的可接受颜色。我们证明，尽管$(γ,μ)$-着色在多个结构良好的图类上仍是NP完全的，但其$k$限制版本对于任意固定的$k$均存在多项式时间解法。通过扩展此形式化定义，我们提出了$k$-$(γ,μ)$-可选性概念，并分析了其表达能力与计算极限。研究结果表明，在区间约束下可接受的列表分配数量显著减少，为经典可选性提供了更具可解性的替代方案，尽管其一般判定问题仍位于多项式层次结构的高层。总体而言，本文通过结构归约为列表着色变体提供了统一视角，为基于区间的模型建立了新的复杂度界限，并揭示了固定尺寸连续颜色范围约束所带来的算法优势。