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

TL;DR

This paper explores the complexity and structural properties of list coloring problems with interval constraints, demonstrating how fixed-size color ranges can simplify computational challenges in graph coloring.

Contribution

It introduces the interval-restricted $k$-$( extgamma, extmu)$-coloring model, proving polynomial-time solvability for fixed $k$ and establishing complexity bounds for interval-based list coloring variants.

Findings

Interval-restricted $k$-$( extgamma, extmu)$-coloring is polynomial-time solvable for fixed $k$.

Complexity results transfer among list coloring variants on certain graph classes.

Interval constraints reduce the number of admissible list assignments, improving tractability.

Abstract

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, $\mu$-coloring, and $(\gamma,\mu)$-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 $k$-$(\gamma,\mu)$-coloring model, where each vertex receives an interval of exactly $k$ consecutive admissible colors. We prove that, although $(\gamma,\mu)$-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 $k$-$(\gamma,\mu)$-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.