### Abstract

An edge of a graph is light when the sum of the degrees of its end-vertices is at most 13. The well-known Kotzig theorem states that every 3-connected planar graph contains a light edge. Later, Borodin [J. Reine Angew. Math., 394 (1989), pp. 180-185] extended this result to the class of planar graphs of minimum degree at least 3. We deal with generalizations of these results for planar graphs of minimum degree 2. Borodin, Kostochka, and Woodall [J. Combin. Theory Ser. B, 71 (1997), pp. 184-204] showed that each such graph contains a light edge or a member of two infinite sets of configurations, called 2-alternating cycles and 3-alternators. This implies that planar graphs with maximum degree Δ > 12 are Δ-edge-choosable. We prove a similar result with 2-alternating cycles and 3-alternators replaced by five fixed bounded-sized configurations called crowns. This gives another proof of Δ-edge-choosability of planar graphs with Δ > 12. However, we show efficient choosability; i.e., we describe a linear-time algorithm for max{Δ, 12}-edge-list-coloring planar graphs. This extends the result of Chrobak and Yung [J. Algorithms, 10 (1989), pp. 35-51].

Original language | English (US) |
---|---|

Pages (from-to) | 93-106 |

Number of pages | 14 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 21 |

Issue number | 1 |

DOIs | |

State | Published - 2007 |

### Fingerprint

### Keywords

- Algorithm
- Choosability
- Kotzig's theorem
- Light edge
- List-coloring
- Planar graph

### ASJC Scopus subject areas

- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

*SIAM Journal on Discrete Mathematics*,

*21*(1), 93-106. https://doi.org/10.1137/050646196

**A generalization of Kotzig's theorem and its application.** / Cole, Richard; Kowalik, Lukasz; Škrekovski, Riste.

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 21, no. 1, pp. 93-106. https://doi.org/10.1137/050646196

}

TY - JOUR

T1 - A generalization of Kotzig's theorem and its application

AU - Cole, Richard

AU - Kowalik, Lukasz

AU - Škrekovski, Riste

PY - 2007

Y1 - 2007

N2 - An edge of a graph is light when the sum of the degrees of its end-vertices is at most 13. The well-known Kotzig theorem states that every 3-connected planar graph contains a light edge. Later, Borodin [J. Reine Angew. Math., 394 (1989), pp. 180-185] extended this result to the class of planar graphs of minimum degree at least 3. We deal with generalizations of these results for planar graphs of minimum degree 2. Borodin, Kostochka, and Woodall [J. Combin. Theory Ser. B, 71 (1997), pp. 184-204] showed that each such graph contains a light edge or a member of two infinite sets of configurations, called 2-alternating cycles and 3-alternators. This implies that planar graphs with maximum degree Δ > 12 are Δ-edge-choosable. We prove a similar result with 2-alternating cycles and 3-alternators replaced by five fixed bounded-sized configurations called crowns. This gives another proof of Δ-edge-choosability of planar graphs with Δ > 12. However, we show efficient choosability; i.e., we describe a linear-time algorithm for max{Δ, 12}-edge-list-coloring planar graphs. This extends the result of Chrobak and Yung [J. Algorithms, 10 (1989), pp. 35-51].

AB - An edge of a graph is light when the sum of the degrees of its end-vertices is at most 13. The well-known Kotzig theorem states that every 3-connected planar graph contains a light edge. Later, Borodin [J. Reine Angew. Math., 394 (1989), pp. 180-185] extended this result to the class of planar graphs of minimum degree at least 3. We deal with generalizations of these results for planar graphs of minimum degree 2. Borodin, Kostochka, and Woodall [J. Combin. Theory Ser. B, 71 (1997), pp. 184-204] showed that each such graph contains a light edge or a member of two infinite sets of configurations, called 2-alternating cycles and 3-alternators. This implies that planar graphs with maximum degree Δ > 12 are Δ-edge-choosable. We prove a similar result with 2-alternating cycles and 3-alternators replaced by five fixed bounded-sized configurations called crowns. This gives another proof of Δ-edge-choosability of planar graphs with Δ > 12. However, we show efficient choosability; i.e., we describe a linear-time algorithm for max{Δ, 12}-edge-list-coloring planar graphs. This extends the result of Chrobak and Yung [J. Algorithms, 10 (1989), pp. 35-51].

KW - Algorithm

KW - Choosability

KW - Kotzig's theorem

KW - Light edge

KW - List-coloring

KW - Planar graph

UR - http://www.scopus.com/inward/record.url?scp=40449093280&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=40449093280&partnerID=8YFLogxK

U2 - 10.1137/050646196

DO - 10.1137/050646196

M3 - Article

AN - SCOPUS:40449093280

VL - 21

SP - 93

EP - 106

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 1

ER -