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

TL;DR

The paper investigates the complexity of list-based graph coloring under interval constraints, proposing an analogy framework to transfer hardness and tractability across List Coloring, μ-coloring, and (γ,μ)-coloring on graph classes closed under pendant-vertex extensions. It introduces the interval-restricted k-(γ,μ)-coloring and shows that for fixed k this variant is polynomial-time solvable on several classical graph classes, while general (γ,μ)-coloring remains NP-complete. It further defines k-(γ,μ)-choosability and proves that, despite interval restriction, the associated decision problem lies at the second level of the polynomial hierarchy (Pi_2^P), though the reduced search space enables practical exact algorithms. Two explicit algorithms are provided for deciding k-(γ,μ)-choosability, illustrating the computational burden of universal quantification over interval lists. Overall, the work offers a unified view of constrained coloring variants via structural reductions and highlights the algorithmic benefits of fixed-size consecutive color ranges.

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, $μ$-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.

Figures (6)

• Figure 1: Example of list coloring applied to processor assignment. Each operation (vertex) has a list of allowed processors (colors), and the graph encodes execution conflicts. The final coloring respects both conflict constraints and individual compatibility restrictions.
• Figure 2: Illustration of the $\mu$-coloring model applied to a channel assignment scenario. Each vertex represents a client with a maximum acceptable channel index $\mu(v)$, and edges denote conflicts that require distinct channel assignments. The final coloring ensures that each client receives a suitable channel while avoiding interference with neighbors.
• Figure 3: Modeling of a real-time task schedule as a $(\gamma, \mu)$-coloring problem. On the left, each horizontal bar represents the allowed execution window $[\gamma(v), \mu(v)]$ of a task, and the colored dot indicates the assigned time $f(v)$. On the right, each vertex in the conflict graph represents a task, with its color matching the scheduled time. Edges represent resource conflicts — i.e., tasks that cannot run at the same time. The interval $[\gamma(v), \mu(v)]$ for each task is also shown next to its corresponding vertex. This coloring ensures a valid and conflict-free schedule within the specified temporal constraints.
• Figure 4: Example of a list coloring on a graph. On the left, each vertex is associated with a list of allowed colors $L(v)$, shown next to the vertex. On the right, a proper list coloring is shown, where each vertex $v$ is assigned a color $f(v) \in L(v)$ such that no two adjacent vertices share the same color. The assigned colors are consistent with the lists and respect the coloring constraint for adjacent vertices.
• Figure 5: An example of a $(\gamma, \mu)$-coloring of a graph. On the left, each vertex is associated with an interval $[\gamma(v), \mu(v)]$ indicating the minimum and maximum allowed colors for that vertex. On the right, a valid coloring is presented in which each vertex $v$ is assigned a color $f(v)$ such that $\gamma(v) \leq f(v) \leq \mu(v)$ and adjacent vertices receive different colors. The vertex colors are shown numerically and visually.

Theorems & Definitions (43)

• Definition 1: erdos:1979
• Definition 2: bonomo2005between
• Definition 3: bonomo2009exploring
• Definition 4: chartrand2019chromatic
• Definition 5: chartrand2019chromatic
• Definition 6: fellows2015tractability
• Lemma 1: gama2019aspects
• proof
• Corollary 1
• proof