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On light cycles in plane triangulations. (English) Zbl 0936.05065
A graph \(H\) is said to be light in the class of graphs \({\mathcal G}\) if there exists a positive integer \(k\) such that each graph \(G\in{\mathcal G}\) that contains an isomorphic copy of \(H\) contains a subgraph \(K\) isomorphic to \(H\) that satisfies the inequality \(\deg_G(v)\leq k\), for all vertices \(v\) of \(K\). The smallest positive integer \(k\) with this property is denoted by \(\varphi(H,{\mathcal G})\).
In the main result of the paper, the authors present a complete classification of cycles \(C_r\) that are light in the class \({\mathcal T}(5)\) of plane triangulations with minimum degree 5, namely, they show that a cycle \(C_r\) is light in \({\mathcal T}(5)\) if and only if \(r\in\{3, 4, 5, 6, 7, 8, 9, 10\}\). As for the numbers \(\varphi(C_r,{\mathcal T}(5))\), they show that \(10\leq \varphi(C_6,{\mathcal T}(5))\leq 11\), \(15\leq\varphi(C_7,{\mathcal T}(5))\leq 17\), \(15\leq \varphi(C_8,{\mathcal T}(5))\leq 29\), \(19\leq \varphi(C_9,{\mathcal T}(5))\leq 41\), \(20\leq \varphi(C_{10},{\mathcal T}(5))\leq 415\) (the remaining three identities are \(\varphi(C_3,{\mathcal T}(5))= 7\), \(\varphi(C_4,{\mathcal T}(5))= 10\), \(\varphi(C_5,{\mathcal T}(5))= 10\); the last two have been shown by S. Jendrol’ and T. Madaras [Discuss. Math., Graph Theory 16, No. 2, 207-217 (1996; Zbl 0877.05050)]; \(C_{10}\) has been shown to be light in \({\mathcal T}(5)\) by T. Madaras ans R. Soták).
Most of the paper is devoted to proving the theorem. The proofs of the fact that none of the \(C_r\)’s with \(r>10\) is light in \({\mathcal T}(5)\) and of the lower bounds are constructive. The upper bounds are the result of a clever application of a “discharge method” to the hypothetical counterexamples.

MSC:
05C38 Paths and cycles
52B10 Three-dimensional polytopes
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[1] Ando, K.; Iwasaki, S.; Kaneko, A., Every 3-connected planar graph has a connected subgraph with small degree sum, (), (in Japanese)
[2] Borodin, O.V., Solution of problems of kotzig and grünbaum concerning the isolation of cycles in planar graphs, Mat. zametki, 46, 5, 9-12, (1989) · Zbl 0694.05027
[3] Borodin, O.V., On the total coloring of plane graphs, J. reine angew. math., 394, 180-185, (1989) · Zbl 0653.05029
[4] Borodin, O.V., Triangles with restricted degree sum of their boundary vertices in plane graphs, Discrete math., 137, 45-51, (1995) · Zbl 0814.05030
[5] Borodin, O.V., Minimal vertex degree sum of a 3-path in plane maps, Discuss. math. graph theory, 17, 279-284, (1997) · Zbl 0906.05017
[6] Borodin, O.V.; Sanders, D.P., On light edges and triangles in planar graphs of minimum degree five, Math. nachr., 170, 19-24, (1994) · Zbl 0813.05020
[7] O.V. Borodin, D.R. Woodall, Short cycles of low weight in normal plane maps with minimum degree, Discuss. Math. Graph Theory, to appear. · Zbl 0927.05069
[8] Fabrici, I.; Jendrol’, S., Subgraphs with restricted degrees of their vertices in planar 3-connected graphs, Graphs combin., 13, 245-250, (1997) · Zbl 0891.05025
[9] I. Fabrici, E. Hexel, S. Jendrol’, H. Walther, On vertex-degree restricted paths in polyhedral graphs, Discrete Math., submitted. · Zbl 0946.05047
[10] Franklin, P., The four colour problem, (), 44, 1737-1936, (1922), or in
[11] Grünbaum, B., New views on some old questions of combinatorial geometry, (), 451-468
[12] Grünbaum, B., Polytopal graphs, MAA studies math., vol. 12, 201-224, (1975) · Zbl 0323.05104
[13] S. Jendrol’, Paths with restricted degrees of their vertices in planar graphs, Czechoslovak Math. J., to appear. · Zbl 1003.05055
[14] Jendrol’, S.; Madaras, T., On light subgraphs in plane graphs of minimum degree five, Discuss. math. graph theory, 16, 207-217, (1996) · Zbl 0877.05050
[15] Kotzig, A., Contribution to the theory of Eulerian polyhedra, Mat.-fyz. čas. SAV (math. slovaca), 5, 111-113, (1955), (in Slovak)
[16] Kotzig, A., On the theory of Euler’s polyhedra, Mat.-fyz. čas. SAV (math. slovaca), 13, 20-31, (1963), (in Russian) · Zbl 0134.19601
[17] Lebesgue, H., Quelques consequences simples de la formule d’Euler, J. math. pures appl., 19, 19-43, (1940) · JFM 66.0736.03
[18] T. Madaras, R. Soták, The 10-cycle C10 is light in the family of all plane triangulations with minimum degree five, Tatra Mt. Math. Publ., to appear.
[19] Wernicke, P., Über den kartographischen vierfarbensatz, Math. ann., 58, 413-426, (1904) · JFM 35.0511.01
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