Light paths and edges in families of outer-1-planar graphs

It is proved that (1) every maximal outer-l-planar graph of order at least k contains a path on k-vertices with all vertices of degree at most 2k + 1 (being sharp for k <= 3), and a path on k-vertices with degree sum at most 5k 1, and further, (2) every maximal outer 1-planar graph contains an edge xy with d(x) + d(y) <= 7, and every outer-1-planar graph with minimum degree at least 2 contains an edge xy with d(x)+d(y) <= 9. Here the bounds 7 and 9 are sharp. (C) 2018 Elsevier B.V. All rights reserved.