On a problem of J. Zaks concerning 5-valent 3-connected planar graphs
The author presents an affirmative solution of the following Eberhard- type problem, posed by J. Zaks: Let \((p_ 5,p_ 6,...)\) be a finite sequence of nonnegative integers; does there exist a 5-valent 3-connected planar graph G such that it has exactly \(p_ k\) k-gons for all \(k\geq 5\), \(m_ i\) of its vertices meet exactly i triangles, \(i=4,5\), and \(m_ 4+2m_ 5=24+3\sum_{k\geq 5}(k-4)p_ k?\) Moreover, some interesting relationships among certain sets of r-valent s-connected graphs embeddable in an orientable surface of genus g (for suitable triples (r,s,g)) are established.
scientific article; zbMATH DE number 3855141