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

Abstract 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 2 k + 1 (being sharp for k ≤ 3 ), and a path on k-vertices with degree sum at most 5 k − 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.

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