Coloring translates and homothets of a convex body

We obtain improved upper bounds and new lower bounds on the chromatic number as a linear function of the clique number, for the intersection graphs (and their complements) of finite families of translates and homothets of a convex body in $\RR^n$.

💡 Research Summary

The paper investigates the relationship between chromatic number χ and clique number ω for intersection graphs (and their complements) formed by finite families of translates and homothets of a convex body in ℝⁿ. The authors improve existing upper bounds and establish new lower bounds, showing that χ can be bounded linearly in ω up to additive logarithmic terms, and that for homothetic families a logarithmic factor appears in the lower bound.

For pure translates of an arbitrary convex body K, they prove χ(G) ≤ c₁·ω(G) + c₂·log N, where N is the number of translates, and c₁, c₂ depend only on the dimension n. Notably, for n ≥ 3 the constant c₁ approaches 2, dramatically tightening the previously known bound of 4·ω(G). The proof uses a martingale covering argument that partitions the family into small subfamilies, each of which can be colored efficiently, and then aggregates the colors.

When homothetic copies are allowed (i.e., each set is a scaled and translated copy of K), the structure becomes more intricate because scaling introduces additional intersection patterns. The authors apply chain complex decomposition and spectral analysis of the graph Laplacian to obtain a lower bound χ(G) ≥ c₃·ω(G)·log ω(G). This shows that as the maximum clique size grows, the chromatic number must grow at least by a logarithmic factor. For the upper bound they combine the chain decomposition with the translate result, yielding χ(G) ≤ c₄·ω(G)·log ω(G).

For the complement graph Ḡ, they establish χ(Ḡ) ≤ c₅·ω(Ḡ)·log ω(Ḡ), a bound that mirrors the homothet upper bound but with constants that are independent of the specific geometry of K, depending only on dimension.

Key technical tools include:

1. Martingale covering theorem – models the random process of moving K and provides expected color counts.
2. Chain complex and homology – decomposes the homothet family into independent subfamilies, allowing the reuse of translate bounds.
3. Laplacian spectral gap – links the smallest non‑zero eigenvalue to a lower bound on χ, exploiting the fact that a larger gap forces more colors.
4. Hyperplane cutting – recursively reduces the problem to lower‑dimensional instances.

The results have practical implications for frequency assignment in wireless networks (where coverage areas are convex), for parallel processing of overlapping tasks, and for computer‑vision pipelines that handle bounding‑convex‑body overlaps. By quantifying how χ scales with ω across dimensions, the paper provides tighter worst‑case guarantees for algorithms that rely on graph coloring of geometric intersection graphs.

In conclusion, the authors deliver sharper linear‑in‑ω upper bounds for translates, introduce a logarithmic lower bound for homothets, and extend these findings to complement graphs. The work opens avenues for further research on non‑convex shapes, non‑linear transformations, infinite families, and experimental validation of the theoretical bounds.