Analogy between List Coloring Problems and the Interval $k$-$(γ,μ)$-choosability property: theoretical aspects of complexity

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.

💡 Research Summary

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The paper establishes a unified framework that connects three well‑studied variants of graph coloring—List Coloring, μ‑coloring, and (γ, μ)‑coloring—through the notions of “analogous” and “p‑analogous” problems. The authors prove that when a graph class is closed under pendant‑vertex extensions, there exist polynomial‑time reductions among these three models that preserve computational complexity. Consequently, any hardness or tractability result known for List Coloring on a particular class (for example, planar graphs, bounded‑degree graphs, or chordal graphs) immediately transfers to μ‑coloring and (γ, μ)‑coloring on the same class. This structural correspondence provides a powerful tool for propagating complexity results across different coloring paradigms.

Building on this foundation, the authors introduce a new model called interval‑restricted k‑(γ, μ)‑coloring. In this model each vertex v receives a list that is exactly k consecutive admissible colors, i.e., the list is an interval