Discussiones Mathematicae Graph Theory  17(2) (1997)  279-284
doi: 10.7151/dmgt.1055

[BIBTex] [PDF]

MINIMAL VERTEX DEGREE SUM OF A 3-PATH IN PLANE MAPS

O.V. Borodin

Novosibirsk State University
Novosibirsk, 630090, Russia

Abstract

Let wk be the minimum degree sum of a path on k vertices in a graph. We prove for normal plane maps that: (1) if w2 = 6, then w3 may be arbitrarily big, (2) if w2 >6, then either w3 ≤ 18 or there is a  ≤ 15-vertex adjacent to two 3-vertices, and (3) if w2 > 7, then w3 ≤ 17.

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

1991 Mathematics Subject Classification: 05C10.

References

[1] O.V. Borodin, Solution of Kotzig's and Grünbaum's problems on the separability of a cycle in plane graph, (in Russian), Matem. zametki 48 (6) (1989) 9-12.
[2] O.V. Borodin, Triangulated 3-polytopes without faces of low weight, submitted.
[3] H. Enomoto and K. Ota, Properties of 3-connected graphs, preprint (April 21, 1994).
[4] K. Ando, S. Iwasaki and A. Kaneko, Every 3-connected planar graph has a connected subgraph with small degree sum I, II (in Japanese), Annual Meeting of Mathematical Society of Japan, 1993.
[5] Ph. Franklin, The four colour problem, Amer. J. Math. 44 (1922) 225-236, doi: 10.2307/2370527.
[6] S. Jendrol', Paths with restricted degrees of their vertices in planar graphs, submitted.
[7] S. Jendrol', A structural property of 3-connected planar graphs, submitted.
[8] A. Kotzig, Contribution to the theory of Eulerian polyhedra, (in Russian), Mat. Cas. 5 (1955) 101-103.