# Minimal vertex degree sum of a 3-path in plane maps

Discussiones Mathematicae Graph Theory (1997)

- Volume: 17, Issue: 2, page 279-284
- ISSN: 2083-5892

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topO.V. Borodin. "Minimal vertex degree sum of a 3-path in plane maps." Discussiones Mathematicae Graph Theory 17.2 (1997): 279-284. <http://eudml.org/doc/270447>.

@article{O1997,

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.},

author = {O.V. Borodin},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {planar graph; structure; degree; path; weight; plane map; minimum degree sum},

language = {eng},

number = {2},

pages = {279-284},

title = {Minimal vertex degree sum of a 3-path in plane maps},

url = {http://eudml.org/doc/270447},

volume = {17},

year = {1997},

}

TY - JOUR

AU - O.V. Borodin

TI - Minimal vertex degree sum of a 3-path in plane maps

JO - Discussiones Mathematicae Graph Theory

PY - 1997

VL - 17

IS - 2

SP - 279

EP - 284

AB - 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.

LA - eng

KW - planar graph; structure; degree; path; weight; plane map; minimum degree sum

UR - http://eudml.org/doc/270447

ER -

## References

top- [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. Zbl0956.52010
- [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. Zbl48.0664.02
- [6] S. Jendrol', Paths with restricted degrees of their vertices in planar graphs, submitted. Zbl1003.05055
- [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. Čas. 5 (1955) 101-103.

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