Describing short paths in plane graphs of girth at least 5

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∈{7,8,9,10}, and by pair (2,2) if g≥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,∞,2), (2,2,6), (2,3,5), (2,4,4), or (3,3,3) if g=5, by one of the triplets (2,2,∞), (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∈{8,9}, by one of the triplets (2,2,3) or (2,3,2) if g≥10, and by the triplet (2,2,2) if g≥16.