Note on the weight of paths in plane triangulations of minimum degree 4 and 5

• Autorzy: Tomás Madaras
• Języki publikacji: EN

Abstrakty

EN

The weight of a path in a graph is defined to be the sum of degrees of its vertices in entire graph. It is proved that each plane triangulation of minimum degree 5 contains a path P₅ on 5 vertices of weight at most 29, the bound being precise, and each plane triangulation of minimum degree 4 contains a path P₄ on 4 vertices of weight at most 31.

Słowa kluczowe

EN

weight of path; plane graph; triangulation

Kategorie tematyczne

• 05C10: Planar graphs; geometric and topological aspects of graph theory
• 05C38: Paths and cycles
• 52B10: Three-dimensional polytopes

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2000
• Tom: 20
• Numer: 2
• Strony: 173-180

Daty

wydano

2000

otrzymano

1999-04-14

poprawiono

2000-08-20

Twórcy

autor

Tomás Madaras

• Department of Geometry and Algebra, P.J. Šafárik University, Jesenná 5, 041 54 Košice, Slovak Republic

Bibliografia

• [1] K. Ando, S. Iwasaki and A. Kaneko, Every 3-connected planar graph has a connected subgraph with small degree sum, Annual Meeting of the Mathematical Society of Japan, 1993 (in Japanese).
• [2] O.V. Borodin, Solution of problems of Kotzig and Grünbaum concerning the isolation of cycles in planar graphs, Mat. Zametki 46 (5) (1989) 9-12.
• [3] O.V. Borodin, Minimal vertex degree sum of a 3-path in plane maps, Discuss. Math. Graph Theory 17 (1997) 279-284, doi: 10.7151/dmgt.1055.
• [4] O.V. Borodin and D.R. Woodall, Short cycles of low weight in normal plane maps with minimum degree 5, Discuss. Math. Graph Theory 18 (1998) 159-164, doi: 10.7151/dmgt.1071.
• [5] I. Fabrici and S. Jendrol', Subgraphs with restricted degrees of their vertices in planar 3-connected graphs, Graphs and Combinatorics 13 (1997) 245-250.
• [6] I. Fabrici and S. Jendrol', Subgraphs with restricted degrees of their vertices in planar graphs, Discrete Math. 191 (1998) 83-90, doi: 10.1016/S0012-365X(98)00095-8.
• [7] I. Fabrici, E. Hexel, S. Jendrol' and H. Walther, On vertex-degree restricted paths in polyhedral graphs, Discrete Math. 212 (2000) 61-73, doi: 10.1016/S0012-365X(99)00209-5.
• [8] P. Franklin, The four color problem, Amer. J. Math. 44 (1922) 225-236, doi: 10.2307/2370527.
• [9] S. Jendrol' and T. Madaras, On light subgraphs in plane graphs of minimum degree five, Discuss. Math. Graph Theory 16 (1996) 207-217, doi: 10.7151/dmgt.1035.
• [10] E. Jucovic, Convex polytopes (Veda, Bratislava, 1981).
• [11] T. Madaras, Note on weights of paths in polyhedral graphs, Discrete Math. 203 (1999) 267-269, doi: 10.1016/S0012-365X(99)00052-7.
• [12] B. Mohar, Light paths in 4-connected graphs in the plane and other surfaces, J. Graph Theory 34 (2000) 170-179, doi: 10.1002/1097-0118(200006)34:2<170::AID-JGT6>3.0.CO;2-P
• [13] P. Wernicke, Über den kartographischen Vierfarbensatz, Math. Ann. 58 (1904) 413-426, doi: 10.1007/BF01444968.

Identyfikatory

DOI

10.7151/dmgt.1117