Light graphs in families of outerplanar graphs

We prove that every 2-connected outerplanar graph of order at least k (k >= 3) contains a path on k vertices with all vertices of degree at most k + 3 and a path on k vertices with degree sum at most 4k - 2. Further, every 2-connected outerplanar graph without adjacent vertices with degree sum <= 5 contains a triangle C(3) with vertices of degrees 2, 4 and b, 4 <= b <= 6, and a 3-star K(1,3) with central vertex of degree 4 and remaining vertices of degrees 2, 2 and b, 4 <= b <= 6, Moreover, all bounds are best possible. (c) 2006 Elsevier B.V. All rights reserved.