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

It is proved that (1) every maximal outer-1-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.