Deza graphs and regular polyhedra

TL;DR

This work classifies regular planar polyhedra by the set of possible numbers of common neighbours $A(G)$ for pairs of distinct vertices, yielding a complete split into two main families: cubic polyhedra with no quadrangular faces and quartic polyhedra with no $4$-cycles. It shows that quartic Deza graphs are precisely the medial graphs $\mathcal M(\Gamma)$ of polyhedra $\Gamma$ with no degree-$4$ vertices, no quadrangular faces, and no degree-$3$ vertex on a triangular face, and it derives tight extremal bounds for the number of triangular faces in terms of the total number of faces; the upper extremal case corresponds to line graphs of cubic polyhedra of girth $5$, while the lower extremal case is realized by medial graphs of specially constrained polyhedra. The paper also fully characterizes $4$-regular polyhedra of type $\{0,1,2,3\}$, showing they either contain a square pyramid or arise from a TT-construction, and never arise as medial graphs. Collectively, these results connect Deza graph theory with the fine structure of regular planar graphs and polyhedra, and they illuminate general properties of regular planar graphs through the lens of common-neighbour counts.

Abstract

We classify all regular polyhedra according to their type i.e., the collection of numbers of common neighbours that any pair of distinct vertices may have (polyhedra are planar, $3$-connected graphs). As an application, we recover the classification of planar Deza graphs. Next, we focus on the class of quartic polyhedral Deza graphs, and completely characterise it in terms of medial graphs of certain specific cubic polyhedra. Furthermore, within the aforementioned class of quartic polyhedral Deza graphs, we study the extremal graphs with respect to the ratio of number of triangular faces to the total. In the maximal extreme, these notably coincide with the class of line graphs of cubic polyhedra of girth $5$. We also fully characterise the quartic polyhedra of type $\{0,1,2,3\}$, and in particular we prove that none of them are medial graphs. On one hand our findings fit within the novel research area of common neighbours in graphs. On the other hand, our findings imply general properties of regular planar graphs and regular polyhedra.

Figures (15)

• Figure 1: The five exceptional polyhedral Deza graphs.
• Figure 2: Illustration of $\mathcal{T}(G_1,G_2)$, where $G_1,G_2$ are the third and second graphs in Figure \ref{['fig:s4']}. Dashed edges are deleted.
• Figure 3: $N(u,v)=\{a,b,c,d\}$, in this cyclic order around $u$.
• Figure 4: Transformations to construct all $3$-connected quadrangulations. A half-edge indicates an edge that is necessarily there, while a triangle on a vertex indicates one or more edges that might be there.
• Figure 5: Assuming that $G$ contains a separating $4$-cycle leads to a contradiction.

Theorems & Definitions (24)

• Proposition 1.1: cf. limaye2005regularbrouwer2006classificationgoryainov2021enumeration
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 1.5
• Definition 1.6
• Proposition 1.7
• Corollary 1.8
• Lemma 2.1
• proof