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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
##### Keywords:
light cycles; light subgraphs; plane triangulations
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##### References:
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