Adjacency Graphs of Polyhedral Surfaces

TL;DR

This work investigates when a graph can be realized as the adjacency graph of polygonal cells on a polyhedral surface in R^3. It proves that any graph can be realized with arbitrary nonconvex polygonal cells, but convex realizability imposes strong constraints, ruling out graphs containing K_5, K_{5,81}, or nonplanar 3-trees, while many nonplanar graphs (e.g., K_{4,4}, K_{3,5}) are realizable with convex cells. The paper also establishes density bounds for realizable graphs, showing e_max(n) is at least Ω(n log n) due to hypercubes and at most O(n^{9/5}) due to forbidding K_{5,81}, with additional constructions achieving high average degree. These results illuminate the trade-off between convexity and density in 3D contact representations and connect to broader questions about embedding, coloring, and graph realizability of higher-genus polyhedral surfaces.

Abstract

We study whether a given graph can be realized as an adjacency graph of the polygonal cells of a polyhedral surface in $\mathbb{R}^3$. We show that every graph is realizable as a polyhedral surface with arbitrary polygonal cells, and that this is not true if we require the cells to be convex. In particular, if the given graph contains $K_5$, $K_{5,81}$, or any nonplanar $3$-tree as a subgraph, no such realization exists. On the other hand, all planar graphs, $K_{4,4}$, and $K_{3,5}$ can be realized with convex cells. The same holds for any subdivision of any graph where each edge is subdivided at least once, and, by a result from McMullen et al. (1983), for any hypercube. Our results have implications on the maximum density of graphs describing polyhedral surfaces with convex cells: The realizability of hypercubes shows that the maximum number of edges over all realizable $n$-vertex graphs is in $Ω(n \log n)$. From the non-realizability of $K_{5,81}$, we obtain that any realizable $n$-vertex graph has $O(n^{9/5})$ edges. As such, these graphs can be considerably denser than planar graphs, but not arbitrarily dense.

Figures (8)

• Figure 1: A convex-polyhedral surface $\mathcal{S}$ and its nonplanar 3-degenerate adjacency graph $\mathcal{G}\xspace(\mathcal{S}\xspace)$.
• Figure 2: A realization of $K_5$ by arbitrary polygons with side contacts in $\mathbb{R}^3$.
• Figure 6: Additional views of the realization of $K_{4,4}$.
• Figure 10: Illustration of Case 2 for the proof of \ref{['lem:oneside']}. (a) A single octant (towards the viewer) with three truncated red polygons and a possible blue polygon within the octant that has a side contact with all three red polygons, (b) the trace of a possible blue polygon forming a triangle, (c) the trace of a possible blue polygon consisting of two rays and a segment, and (d) two traces yield at least one blue polygon that is not one-sided.
• Figure 11: The unique minimal nonplanar 3-tree, which we call triple-stacked triangle.

Theorems & Definitions (18)

• Proposition 1
• Corollary 2
• Proposition 3
• Proposition 4
• Lemma 5
• Proposition 6: thomassenColorCriticalGraphs
• Theorem 7
• Proposition 8
• Proposition 9
• Theorem 10