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Minimal vertex degree sum of a 3-path in plane maps. (English) Zbl 0906.05017
A plane map is normal if every face and every vertex of the map is incident with at least three edges. Let $$w_k$$ be the minimum degree sum, in a graph $$G$$, of the vertices on a path of order $$k$$ in $$G$$. The author shows that, for normal plane maps: (1) If $$w_2= 6$$, then $$w_3$$ can be arbitrarily large. (2) If $$w_2> 6$$, then either $$w_3\leq 18$$ or there is a vertex of degree at most 15 adjacent to two vertices of degree 3. (3) If $$w_2>7$$, then $$w_3\leq 17$$.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C38 Paths and cycles
##### Keywords:
plane map; minimum degree sum; path
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