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Describing short paths in plane graphs of girth at least 5. (English) Zbl 1302.05040
Summary: We prove that every connected plane graph of given girth $$g$$ and minimum degree at least 2 contains an edge whose degrees are bounded from above by one of the pairs $$(2, 5)$$ or $$(3, 3)$$ if $$g = 5$$, by pair $$(2, 5)$$ if $$g = 6$$, by pair $$(2, 3)$$ if $$g \in \{7, 8, 9, 10 \}$$, and by pair $$(2, 2)$$ if $$g \geq 11$$. Further we prove that every connected plane graph of given girth $$g$$ and minimum degree at least 2 has a path on three vertices whose degrees are bounded from above by one of the triplets $$(2, \infty, 2)$$, $$(2, 2, 6)$$, $$(2, 3, 5)$$, $$(2, 4, 4)$$, or $$(3, 3, 3)$$ if $$g = 5$$, by one of the triplets $$(2, 2, \infty)$$, $$(2, 3, 5)$$, $$(2, 4, 3)$$, or $$(2, 5, 2)$$ if $$g = 6$$, by one of the triplets $$(2, 2, 6)$$, $$(2, 3, 3)$$, or $$(2, 4, 2)$$ if $$g = 7$$, by one of the triplets $$(2, 2, 5)$$ or $$(2, 3, 3)$$ if $$g \in \{8, 9 \}$$, by one of the triplets $$(2, 2, 3)$$ or $$(2, 3, 2)$$ if $$g \geq 10$$, and by the triplet $$(2, 2, 2)$$ if $$g \geq 16$$.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C38 Paths and cycles 05C07 Vertex degrees 05C35 Extremal problems in graph theory 05C22 Signed and weighted graphs 05C40 Connectivity
##### Keywords:
plane graph; structural property; girth; 3-path; weight
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##### References:
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