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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
##### Keywords:
3-polytope; edge path; degree
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