A generalization of Kotzig's theorem and its application

Richard Cole, Lukasz Kowalik, Riste Škrekovski

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)93-106
Number of pages14
JournalSIAM Journal on Discrete Mathematics
Volume21
Issue number1
DOIs
StatePublished - 2007

Fingerprint

Planar graph
Theorem
Choosability
Minimum Degree
Coloring
List Coloring
Cycle
Configuration
Edge Coloring
Linear-time Algorithm
Graph in graph theory
Maximum Degree
Generalization
Connected graph
Imply

Keywords

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

ASJC Scopus subject areas

  • Applied Mathematics
  • Discrete Mathematics and Combinatorics

Cite this

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

In: SIAM Journal on Discrete Mathematics, Vol. 21, No. 1, 2007, p. 93-106.

Research output: Contribution to journalArticle

Cole, Richard ; Kowalik, Lukasz ; Škrekovski, Riste. / A generalization of Kotzig's theorem and its application. In: SIAM Journal on Discrete Mathematics. 2007 ; Vol. 21, No. 1. pp. 93-106.
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