### Abstract

A subgraph of a plane graph is light if each of its vertices has a small degree in the entire graph. Consider the class script T sign(5) of plane triangulations of minimum degree 5. It is known that each G ∈ script T sign(5) contains a light triangle. From a recent result of Jendrol' and Madaras the existence of light cycles C_{4} and C_{5} in each G ∈ script T sign(5) follows. We prove here that each G ∈ script T sign(5) contains also light cycles C_{6},C_{7},C_{8} and C_{9} such that every vertex is of degree at most 11, 17,29 and 41, respectively. Moreover, we prove that no cycle C_{k} with k≥11 is light in the class script T sign(5).

Original language | English |
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Pages (from-to) | 453-467 |

Number of pages | 15 |

Journal | Discrete Mathematics |

Volume | 197-198 |

Publication status | Published - Feb 28 1999 |

### Fingerprint

### Keywords

- Cycles
- Light subgraph
- Planar graph
- Triangulation

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*197-198*, 453-467.

**On light cycles in plane triangulations.** / Jendrol', Stanislav; Madaras, Tomáš; Soták, Roman; Tuza, Z.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 197-198, pp. 453-467.

}

TY - JOUR

T1 - On light cycles in plane triangulations

AU - Jendrol', Stanislav

AU - Madaras, Tomáš

AU - Soták, Roman

AU - Tuza, Z.

PY - 1999/2/28

Y1 - 1999/2/28

N2 - A subgraph of a plane graph is light if each of its vertices has a small degree in the entire graph. Consider the class script T sign(5) of plane triangulations of minimum degree 5. It is known that each G ∈ script T sign(5) contains a light triangle. From a recent result of Jendrol' and Madaras the existence of light cycles C4 and C5 in each G ∈ script T sign(5) follows. We prove here that each G ∈ script T sign(5) contains also light cycles C6,C7,C8 and C9 such that every vertex is of degree at most 11, 17,29 and 41, respectively. Moreover, we prove that no cycle Ck with k≥11 is light in the class script T sign(5).

AB - A subgraph of a plane graph is light if each of its vertices has a small degree in the entire graph. Consider the class script T sign(5) of plane triangulations of minimum degree 5. It is known that each G ∈ script T sign(5) contains a light triangle. From a recent result of Jendrol' and Madaras the existence of light cycles C4 and C5 in each G ∈ script T sign(5) follows. We prove here that each G ∈ script T sign(5) contains also light cycles C6,C7,C8 and C9 such that every vertex is of degree at most 11, 17,29 and 41, respectively. Moreover, we prove that no cycle Ck with k≥11 is light in the class script T sign(5).

KW - Cycles

KW - Light subgraph

KW - Planar graph

KW - Triangulation

UR - http://www.scopus.com/inward/record.url?scp=0039111034&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0039111034&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0039111034

VL - 197-198

SP - 453

EP - 467

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

ER -