[Paper Review] Chromatic numbers for the hyperbolic plane and discrete analogs

Hugo Parlier, Camille Petit|ArXiv.org|Jan 30, 2017

Limits and Structures in Graph Theory2 references3 citations

TL;DR

This paper investigates chromatic numbers for the hyperbolic plane and q-regular trees under distance constraints, introducing a hyperbolic checkerboard coloring strategy to establish linear upper bounds in terms of distance d. It proves that χ(ℍ, d) ≤ 5(⌈d/log(4)⌉ + 1) for d ≥ 2 log(3), and provides tight exponential bounds for interval chromatic numbers, showing χ(ℍ, [d,cd]) grows like e^{(cd−1)/2} up to polynomial factors.

ABSTRACT

We study colorings of the hyperbolic plane, analogously to the Hadwiger-Nelson problem for the Euclidean plane. The idea is to color points using the minimum number of colors such that no two points at distance exactly $d$ are of the same color. The problem depends on $d$ and, following a strategy of Kloeckner, we show linear upper bounds on the necessary number of colors. In parallel, we study the same problem on $q$-regular trees and show analogous results. For both settings, we also consider a variant which consists in replacing $d$ with an interval of distances.

Motivation & Objective

• To determine the minimum number of colors required to color the hyperbolic plane such that no two points at exact distance d share the same color.
• To extend the chromatic number problem to q-regular trees as discrete analogs of the hyperbolic plane.
• To analyze the behavior of chromatic numbers when distances are constrained to an interval [d, cd] rather than a single value.
• To provide improved upper bounds for both the hyperbolic plane and trees using a structured checkerboard coloring method.
• To establish lower bounds via clique constructions and interval clique numbers, particularly for large d.

Proposed method

• Uses a hyperbolic checkerboard coloring strategy inspired by Székely and Kloeckner, partitioning the hyperbolic plane into rectangular strata with width w = d and height h = log(4).
• Colors strata periodically, reusing colors every (⌊cd⌋ + 1) strata to manage interval coloring constraints.
• Applies hyperbolic trigonometry to estimate angular separation between points on a circle of radius (c−1)d/2 to bound the number of mutually d-distant points.
• For trees, uses a horocyclic decomposition and bundles vertices sharing a common ancestor at distance ⌊cd/2⌋ + 1 to group vertices for color assignment.
• Employs super-bundles of vertices with common ancestors within ⌊cd/2⌋ + 1 steps to ensure minimum distance > cd between differently colored bundles.
• Derives lower bounds by constructing large cliques: for ℍ, using points on a circle with pairwise distances in [d, cd]; for T_q, using vertices at fixed distances from a root.

Experimental results

Research questions

• RQ1 What is the asymptotic growth of the chromatic number χ(ℍ, d) as d increases?
• RQ2 How does the chromatic number behave when distances are restricted to an interval [d, cd] rather than a single value?
• RQ3 Can discrete analogs like q-regular trees model the chromatic behavior of the hyperbolic plane?
• RQ4 What are the tightest possible upper and lower bounds for χ(ℍ, d) and χ(T_q, d) for both single and interval distance constraints?
• RQ5 How do the chromatic numbers of the hyperbolic plane and trees compare in terms of growth rate with respect to d and c?

Key findings

• For d ≥ 2 log(3), the chromatic number of the hyperbolic plane satisfies χ(ℍ, d) ≤ 5(⌈d/log(4)⌉ + 1), providing a linear upper bound in d.
• For small d ≤ 2 log(2), the paper establishes a tighter bound: χ(ℍ, d) ≤ 9.
• For the interval chromatic number χ(ℍ, [d, cd]), the paper proves exponential growth: 2e^{(cd−1)/2} < χ(ℍ, [d, cd]) < 2(2e^{(cd−1)/2} + 1)(cd + 1) for sufficiently large d.
• For q-regular trees, the chromatic number χ(T_q, d) is exactly 2 when d is odd, and at most (q−1)(d+1) when d is even.
• The interval chromatic number for trees satisfies q(q−1)^{⌊cd/2⌋−⌈d/2⌉} ≤ χ(T_q, [d, cd]) ≤ (q−1)^{⌊cd/2+1⌋}(⌊cd⌋ + 1), showing exponential growth in cd.
• A generalized Moser spindle construction yields a lower bound of q+1 for χ(T_3, 8), suggesting the chromatic number exceeds the clique number, indicating non-trivial structure.