Discussiones Mathematicae Graph Theory 17(2)
(1997) 279-284

doi: 10.7151/dmgt.1055

O.V. Borodin

*Novosibirsk State University
Novosibirsk, 630090, Russia*

Let w_{k} be the minimum degree sum of a path on k vertices in a graph. We
prove for normal plane maps that: (1) if w_{2} = 6, then w_{3} may be
arbitrarily big, (2) if w_{2} >6, then either w_{3}
≤ 18 or there is a ≤ 15-vertex
adjacent to two 3-vertices, and (3) if w_{2} > 7,
then w_{3} ≤ 17.

**Keywords:** planar graph, structure, degree, path, weight.

**1991 Mathematics Subject Classification:** 05C10.

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