Drawing cubic graphs with the four basic slopes

We show that every cubic graph can be drawn in the plane with straight-line edges using only the four basic slopes ${0,\pi/4,\pi/2,3\pi/4}$. We also prove that four slopes have this property if and only if we can draw $K_4$ with them.

💡 Research Summary

The paper addresses the long‑standing question of how many distinct slopes are needed to draw every cubic (3‑regular) graph with straight‑line edges. The “slope number” sl(G) of a graph G is the smallest number of different edge slopes that suffice for a straight‑line drawing. Trivially sl(G) ≥ ⌈Δ/2⌉ where Δ is the maximum degree, so for cubic graphs the lower bound is 2. Earlier work showed that five slopes always suffice, and that four slopes are enough for connected cubic graphs, but a universal four‑slope construction for all cubic graphs (including disconnected ones) was missing.

The authors prove Theorem 1.1: every cubic graph, regardless of connectivity, can be drawn using only the four basic slopes {0, π/4, π/2, 3π/4} (horizontal, vertical, and the two diagonal directions). The proof proceeds in several stages.

1. Preliminaries and Super‑cycles – A super‑cycle is defined as a connected subgraph where every vertex has degree at least two and not all vertices have degree two. Using a breadth‑first search, Lemma 2.4 guarantees a short cycle of length O(log n) in any n‑vertex cubic graph. Lemma 2.5 extends this to a super‑cycle whose size is bounded by a logarithmic function of n and the girth g.

2. M‑cuts – An M‑cut is a cut whose crossing edges form a matching. Lemma 2.6 shows that if a cubic graph contains a super‑cycle with s vertices and the total number of vertices exceeds 2s − 2, then the graph possesses a suitable M‑cut of size at most s − 2. Such a cut separates the graph into two components, each of which is itself a super‑cycle.

3. Large graphs (n ≥ 18) – By combining Lemmas 2.4–2.6, Corollary 2.7 establishes that every cubic graph with at least 18 vertices indeed has a suitable M‑cut. The two resulting components are subcubic (maximum degree three) and can be drawn with the four basic slopes by invoking Theorem 2.3 from a previous paper, which guarantees a four‑slope drawing for any subcubic graph with prescribed x‑coordinates for degree‑2 vertices. One component is rotated by 180° and shifted vertically so that the edges of the M‑cut become vertical (slope π/2). Because the subcubic drawings guarantee no other vertices lie on those vertical lines, the combined drawing is valid and uses only the four basic slopes.

4. Small graphs (n ≤ 16) – For the remaining small cases the authors use structural lemmas:

   • Lemma 2.9 handles graphs with a cut‑vertex (which forces a bridge, providing a trivial M‑cut).
   • Lemma 2.10 deals with graphs that have a two‑vertex separating set, again yielding an appropriate M‑cut.
   • Lemma 2.11 (due to Max Engelstein) shows that any 3‑connected cubic graph possessing a Hamiltonian cycle can be drawn with the four basic slopes. Consequently, the only graphs left to check are 3‑connected, non‑Hamiltonian cubic graphs with at most 16 vertices. These are finitely many and can be enumerated from known catalogs; the authors verify each by a short computer program (code supplied in the appendix). Thus the theorem holds for all cubic graphs.

5. Characterisation of admissible slope sets (Theorem 1.2) – The authors define a set S of four slopes to be “good” if every cubic graph can be drawn using S. They prove that the following are equivalent: (i) S is good, (ii) S is an affine image of the basic four slopes, (iii) K₄ can be drawn with slopes from S. Hence, the ability to draw K₄ is both necessary and sufficient for a four‑slope set to work for all cubic graphs.

The paper concludes with remarks on open problems: whether the slope number is bounded for graphs of maximum degree four remains unknown; connections to other graph parameters such as thickness, geometric thickness, and planar slope number are discussed. The authors also note related work on RAC (right‑angle crossing) drawings and suggest that their four‑slope construction yields RAC drawings for cubic graphs with at most one bend per edge.

In summary, the authors close a gap in the literature by showing that a fixed, simple set of four slopes suffices for every cubic graph, and they provide a clean geometric characterisation of all four‑slope families that have this universal property. This result deepens our understanding of the interplay between graph degree, drawing aesthetics, and combinatorial constraints.