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A structural property of convex 3-polytopes. (English) Zbl 0893.52007
A \((d_1,d_2, \dots, d_k)\)-path on a convex 3-polytope (3-connected planar graph) is one whose successive vertices have degrees \(d_1\), \(d_2, \dots, d_k\).
The author shows here that a 3-polytope always has \((a,b,c)\)-paths for certain restrictions on the degrees. In particular, generalizing the (best possible) cases \(k=1\) \((a\leq 5)\) and \(k=2(a+b\leq 13\), due to A. Kotzig [Mat.-Fyz. Čas., Slovensk. Akad. Vied 5, 101-103 (1955)]), he shows that \(a+b +c\leq 23\) is attainable; he conjectures that 21 is the correct bound.

MSC:
52B10 Three-dimensional polytopes
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
05C10 Planar graphs; geometric and topological aspects of graph theory
05C38 Paths and cycles
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