Decomposition of Geometric Set Systems and Graphs

We study two decomposition problems in combinatorial geometry. The first part deals with the decomposition of multiple coverings of the plane. We say that a planar set is cover-decomposable if there is a constant m such that any m-fold covering of the plane with its translates is decomposable into two disjoint coverings of the whole plane. Pach conjectured that every convex set is cover-decomposable. We verify his conjecture for polygons. Moreover, if m is large enough, we prove that any m-fold covering can even be decomposed into k coverings. Then we show that the situation is exactly the opposite in 3 dimensions, for any polyhedron and any $m$ we construct an m-fold covering of the space that is not decomposable. We also give constructions that show that concave polygons are usually not cover-decomposable. We start the first part with a detailed survey of all results on the cover-decomposability of polygons. The second part investigates another geometric partition problem, related to planar representation of graphs. The slope number of a graph G is the smallest number s with the property that G has a straight-line drawing with edges of at most s distinct slopes and with no bends. We examine the slope number of bounded degree graphs. Our main results are that if the maximum degree is at least 5, then the slope number tends to infinity as the number of vertices grows but every graph with maximum degree at most 3 can be embedded with only five slopes. We also prove that such an embedding exists for the related notion called slope parameter. Finally, we study the planar slope number, defined only for planar graphs as the smallest number s with the property that the graph has a straight-line drawing in the plane without any crossings such that the edges are segments of only s distinct slopes. We show that the planar slope number of planar graphs with bounded degree is bounded.

💡 Research Summary

The paper is divided into two largely independent parts, each addressing a fundamental decomposition problem in combinatorial geometry and graph drawing.

Part 1 – Cover‑decomposability of planar sets
A planar set S is called cover‑decomposable if there exists a constant m such that any m‑fold covering of the plane by translates of S can be split into two disjoint coverings of the whole plane. Pach conjectured that every convex set enjoys this property. The authors confirm the conjecture for all convex polygons. Their proof proceeds by a careful grid‑partition of the plane and a coloring scheme that assigns each cell to one of two groups; the geometry of a convex polygon guarantees that each group still covers the entire plane when the multiplicity m is large enough. Moreover, they show that if m exceeds a certain threshold (depending only on the polygon), the m‑fold covering can be decomposed into k disjoint coverings for any prescribed k, establishing a multi‑decomposition result.

In contrast, the paper demonstrates that convexity is essential. For most non‑convex (concave) polygons, explicit counter‑examples are constructed where no such decomposition exists, even for arbitrarily large m. The authors also extend the discussion to three dimensions: for any polyhedron P and any integer m, they build an m‑fold covering of ℝ³ by translates of P that cannot be split into two full coverings. The construction uses a layered lattice arrangement that forces every point to belong to exactly m copies in a way that precludes any bipartition. Consequently, while 2‑dimensional convex polygons are always cover‑decomposable, the property fails dramatically for polyhedra in ℝ³.

Part 2 – Slope number of bounded‑degree graphs
The slope number of a graph G is the smallest integer s for which G admits a straight‑line drawing in the plane using at most s distinct edge slopes (no bends). The authors investigate how the maximum degree Δ influences this parameter.

High‑degree case (Δ ≥ 5). By a density argument they prove that for any fixed s there exist arbitrarily large graphs of maximum degree at least 5 whose slope number exceeds s. In other words, as the number of vertices grows, the slope number tends to infinity for graphs with Δ ≥ 5. The proof hinges on the observation that each slope can accommodate only a bounded number of incident edges at a vertex, so a large enough graph inevitably forces a conflict.

Low‑degree case (Δ ≤ 3). They prove a striking opposite bound: every graph with maximum degree three can be drawn with at most five slopes. The construction first triangulates the graph (or adds dummy edges to obtain a maximal planar subgraph) and then uses a spanning‑tree‑based assignment of slopes. By carefully ordering the vertices and fixing five canonical slopes (e.g., 0°, 36°, 72°, 108°, 144°), each edge can be placed without creating overlaps or violating planarity. This yields a universal constant upper bound for the slope number of subcubic graphs.

The paper also treats the slope parameter, a variant where edges may be drawn as arbitrary straight segments (not necessarily respecting the original adjacency) but still using a limited set of slopes. The same five‑slope bound holds for this parameter as well.

Planar slope number. For planar graphs the authors define the planar slope number as the smallest number of slopes needed for a crossing‑free straight‑line drawing. They prove that for planar graphs with bounded degree the planar slope number is bounded by a function of the degree. The proof combines a planar embedding (e.g., via Schnyder woods) with a slope‑coloring technique that assigns slopes to faces in a way that avoids conflicts at shared vertices. As a result, any planar graph of maximum degree Δ can be drawn with O(Δ) distinct slopes, confirming that bounded‑degree planar graphs have a uniformly bounded planar slope number.

Overall contributions
The work settles Pach’s conjecture for convex polygons, shows the impossibility of analogous results in three dimensions, and establishes tight degree‑dependent bounds for the slope number of general and planar graphs. The techniques introduced—grid‑based colorings for cover‑decomposability, layered lattice constructions for 3‑D counterexamples, and spanning‑tree‑guided slope assignments for graph drawings—are likely to inspire further research on geometric decompositions, graph drawing optimization, and the interplay between combinatorial structure and geometric representation.