Planar cubic graphs of small diameter

TL;DR

This paper constructs a family of cubic planar graphs with faces of length at most 7 that have logarithmic diameter, challenging previous beliefs about diameter bounds in such graphs.

Contribution

It introduces a new family of cubic planar graphs with small faces and logarithmic diameter, countering prior conjectures about diameter growth.

Findings

Cubic planar graphs with faces of length at most 7 can have diameter in O(log n)

Counterexample to the suspicion that such graphs have diameter in Omega(√n)

Provides explicit construction of these graphs

Abstract

Cubic planar $n$-vertex graphs with faces of length at most $6$, e.g., fullerene graphs, have diameter in $\Omega(\sqrt{n})$. It has been suspected, that a similar result can be shown for cubic planar graphs with faces of bounded length. This note provides a family of cubic planar $n$-vertex graphs with faces of length at most $7$ and diameter in ${O}(\log n)$, thus refuting the above suspicion.

Full text

A fullerene graph is a cubic $3$-connected planar graph all of whose faces are of length $5$ or $6$. They model fullerene molecules, i.e., molecules consisting only of carbon atoms other than graphite and diamonds. The search for correlations of graph invariants and chemical stability motivates the study of fullerenes in chemical graph theory, see e.g. [2, 4].

The diameter of a planar cubic graph can be logarithmic in the number of vertices. However, the diameter of a fullerene graph on $n$ vertices is at least $\frac{1}{6}\sqrt{24n-15}-\frac{1}{2}$, as proved in [1]. Indeed, the proof of this lower bound extends to planar cubic graphs whose faces are of size at most $6$. It has been suspected, that a similar lower bound holds for the diameter of cubic planar graphs with faces of bounded length, see [1, 2]. This is false as shown by the following family $\{G_{k}\mid k\geq 2\}$.

Let $k\geq 2$. Denote by $T_{k}$ the rooted tree all of whose internal vertices are of degree $3$ and all leaves are at distance $k$ from the root. To define $G_{k}$, first take two copies of $T_{k}$ and glue them by identifying corresponding leaves, see Figure 1. Now, subdivide all edges except those incident to leaves or roots. In Figure 1, the new vertices are black, while the original vertices of the trees are white. Fix a plane embedding of the obtained graph such that both roots are on the outer face. Note that, for each tree the vertices in the tree of a given distance $d$ to its root are naturally ordered from left to right. Note that all the subdivision vertices and all the leaves are exactly the vertices at even distance to the root of their tree. Now, for each of the two trees and for every even integer $d$ with $2\leq d\leq 2k-2$, let $v_{0},\ldots,v_{\ell}$ be the vertices at distance $d$ from the root in the tree ordered from left to right. Note that $\ell$ is odd as $d\geq 2$. Put edge $v_{i}v_{i+1}$ for every even $i$ (where $i+1$ is taken modulo $\ell+1$). This completes the definition of $G_{k}$.

The added edges between leaves and subdivision vertices did not destroy planarity, see Figure 1. After gluing the two trees together and subdividing the edges the only vertices of degree $2$ were the leaves and the subdivision vertices. Each of them got one new incident edge, thus $G_{k}$ is cubic.

The faces of $G_{k}$ have lengths $4,5,6$ or $7$. Indeed, every face contains a tree vertex (white) thus it is enough to analyze faces around all the tree vertices. Let $v$ be a vertex of one of the underlying trees in $G_{k}$. If $v$ is the root, then all faces adjacent to $v$ are of length $5$. If $v$ is an internal vertex, then it is adjacent to a $6$-face and two $7$-faces with an exception that when $v$ is adjacent to leaves of its tree, then one of the $7$-faces degenerates to a $4$-face. If $v$ is a leaf, then it is adjacent to a $4$-face and two $6$-faces. See Figure 2.

Clearly, $|V(G_{k})|=n\geq 2^{k}$. On the other hand, every two vertices in $G_{k}$ can be joined by a path of length at most $3k$. Therefore the diameter of $G_{k}$ is at most $3k\leq 3\log(n)$.

Let us finally mention that in [3] fullerene graphs of diameter $\sqrt{\frac{4n}{3}}$ have been constructed. Maybe this is the smallest diameter a fullerene graph can have.

Acknowledgments

The first author was supported by the Agence nationale de la recherche through project ANR-17-CE40-0015 and by the Ministerio de Economía, Industria y Competitividad through grant RYC-2017-22701.