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The 10-cycle $$C_{10}$$ is light in the family of all plane triangulations with minimum degree five. (English) Zbl 0951.05031
A subgraph of a plane graph is light if each of its vertices has a small degree in the entire graph. Consider the class $$\mathcal T(5)$$ of all plane triangulations of minimum degree 5. It is known that each $$G\in \mathcal T(5)$$ contains a light triangle, see, e.g., O. V. Borodin [Solution of Kotzig-Grünbaum problems on separation of a cycle in planar graphs, Mat. Zametki 46, No. 5, 9-12 (1989; Zbl 0694.05027)]. From a recent result of S. Jendrol’ and T. Madaras [Discuss. Math., Graph Theory 16, No. 2, 207-217 (1996; Zbl 0877.05050)], the existence of light cycles $$C_4$$ and $$C_5$$ in each $$G\in \mathcal T(5)$$ follows. S. Jendrol’, T. Madaras, R. Soták and Zs. Tuza [Discrete Math. 197-198, 453-467 (1999; Zbl 0936.05065)] proved that each $$G\in \mathcal T(5)$$ contains light cycles $$C_6, C_7, C_8$$ and $$C_9$$, and that no cycle $$C_k$$ with $$k\geq 11$$ is light in the class $$\mathcal T(5)$$. In this paper the authors supplement the above-mentioned results by proving that each $$G\in \mathcal T(5)$$ also contains a light cycle $$C_{10}$$ such that every of its vertices is of degree at most 415. So the characterization of all light cycles in $$\mathcal T(5)$$ is complete.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C38 Paths and cycles 52B10 Three-dimensional polytopes
##### Keywords:
planar graph; light subgraph; cycles; triangulation