Recursive generation of simple planar 5-regular graphs and pentangulations

Abstract

We describe how the 5-regular simple planar graphs can all be obtained from an elementary family of starting graphs by repeatedly applying a few local expansion operations. The proof uses an amalgam of theory and computation. By incorporating the recursion into the canonical construction path method of isomorph rejection, a generator of non-isomorphic embedded 5-regular planar graphs is obtained with time complexity O(n 2 ) per isomorphism class. A similar result is obtained for simple planar pentangulations with minimum degree 2.

The Weisfeiler-Leman dimension of planar graphs is at most 3

“…We conclude that n = 12. There is only one 5-regular graph on 12 vertices, the icosahedron (see for example [16]).…”

Locally recoverable codes from planar graphs

“…A tetrahedron embedded into the plane contains 4 vertices of degree 3, 6 edges, and 4 faces. The code generated by the tetrahedron is a [6,3] code with locality r = 2 and availability t = 2. The tetrahedron embedded in the plane and one example codeword are shown in Figure 2.…”

“…Case 3: 5-regular planar graphs Families of 5-regular simple planar graphs are generated in [3]. An infinite family D 1 , D 2 , .…”

Matchings in Geometric Graphs

“…In a (strong) matching problem, usually we are looking for a (strong) matching of maximum cardinality in a specific geometric graph. Dillencourt [26] proved that the Delaunay triangulation of P, denoted by DT (P), contains a perfect matching. Ábrego et al [2] proved that DT (P) has a strong circle matching of size at least (n − 1)/8 .…”

“…Figure 8.8 shows a 5-regular geometric graph on a set of 12 points in the plane which contains five non-crossing edge-disjoint plane matchings. In [26], the authors showed how to generate an infinite family of 5-regular planar graphs using the graph in Figure 8 It is obvious that if P contains at least four points, the minimum length Hamiltonian cycle in K(P) contains two non-crossing edge-disjoint plane matchings. Alternatively, in [29] the following result is proved.…”