Zigzag and central circuit structure of \((\{1,2,3\}, 6)\)-spheres. (English) Zbl 1245.05028
Summary: We consider 6-regular plane graphs whose faces have size \(1, 2\) or 3. In Section 2 a practical enumeration method is given that allowed us to enumerate them up to \(53\) vertices. Subsequently, in Section 3 we enumerate all possible symmetry groups of the spheres that showed up. In Section 4 we introduce a new Goldberg-Coxeter construction that takes a 6-regular plane graph \(G_0\), two integers \(k\) and \(l\) and returns two 6-regular plane graphs.
Then in the final section, we consider the notions of zigzags and central circuits for the considered graphs. We introduced the notions of tightness and weak tightness for them and prove an upper bound on the number of zigzags and central circuits of such tight graphs. We also classify the tight and weakly tight graphs with simple zigzags or central circuits.
Then in the final section, we consider the notions of zigzags and central circuits for the considered graphs. We introduced the notions of tightness and weak tightness for them and prove an upper bound on the number of zigzags and central circuits of such tight graphs. We also classify the tight and weakly tight graphs with simple zigzags or central circuits.
MSC:
05C10 | Planar graphs; geometric and topological aspects of graph theory |
05C30 | Enumeration in graph theory |
05E05 | Symmetric functions and generalizations |
Online Encyclopedia of Integer Sequences:
Number of representations of n by the quadratic form x^2 + xy + y^2 with 0 <= x <= y.Number of loopless ({1,2,3},6)-spheres with n vertices.