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Edge-choosability and total-choosability of planar graphs with no adjacent 3-cycles

Daniel W. Cranston

Discussiones Mathematicae Graph Theory (2009)

  • Volume: 29, Issue: 1, page 163-178
  • ISSN: 2083-5892

Abstract

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Let G be a planar graph with no two 3-cycles sharing an edge. We show that if Δ(G) ≥ 9, then χ'ₗ(G) = Δ(G) and χ''ₗ(G) = Δ(G)+1. We also show that if Δ(G) ≥ 6, then χ'ₗ(G) ≤ Δ(G)+1 and if Δ(G) ≥ 7, then χ''ₗ(G) ≤ Δ(G)+2. All of these results extend to graphs in the projective plane and when Δ(G) ≥ 7 the results also extend to graphs in the torus and Klein bottle. This second edge-choosability result improves on work of Wang and Lih and of Zhang and Wu. All of our results use the discharging method to prove structural lemmas about the existence of subgraphs with small degree-sum. For example, we prove that if G is a planar graph with no two 3-cycles sharing an edge and with Δ(G) ≥ 7, then G has an edge uv with d(u) ≤ 4 and d(u)+d(v) ≤ Δ(G)+2. All of our proofs yield linear-time algorithms that produce the desired colorings.

How to cite

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Daniel W. Cranston. "Edge-choosability and total-choosability of planar graphs with no adjacent 3-cycles." Discussiones Mathematicae Graph Theory 29.1 (2009): 163-178. <http://eudml.org/doc/270180>.

@article{DanielW2009,
abstract = {Let G be a planar graph with no two 3-cycles sharing an edge. We show that if Δ(G) ≥ 9, then χ'ₗ(G) = Δ(G) and χ''ₗ(G) = Δ(G)+1. We also show that if Δ(G) ≥ 6, then χ'ₗ(G) ≤ Δ(G)+1 and if Δ(G) ≥ 7, then χ''ₗ(G) ≤ Δ(G)+2. All of these results extend to graphs in the projective plane and when Δ(G) ≥ 7 the results also extend to graphs in the torus and Klein bottle. This second edge-choosability result improves on work of Wang and Lih and of Zhang and Wu. All of our results use the discharging method to prove structural lemmas about the existence of subgraphs with small degree-sum. For example, we prove that if G is a planar graph with no two 3-cycles sharing an edge and with Δ(G) ≥ 7, then G has an edge uv with d(u) ≤ 4 and d(u)+d(v) ≤ Δ(G)+2. All of our proofs yield linear-time algorithms that produce the desired colorings.},
author = {Daniel W. Cranston},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {list coloring; edge coloring; total coloring; Vizing's Conjecture; Vizing's conjecture},
language = {eng},
number = {1},
pages = {163-178},
title = {Edge-choosability and total-choosability of planar graphs with no adjacent 3-cycles},
url = {http://eudml.org/doc/270180},
volume = {29},
year = {2009},
}

TY - JOUR
AU - Daniel W. Cranston
TI - Edge-choosability and total-choosability of planar graphs with no adjacent 3-cycles
JO - Discussiones Mathematicae Graph Theory
PY - 2009
VL - 29
IS - 1
SP - 163
EP - 178
AB - Let G be a planar graph with no two 3-cycles sharing an edge. We show that if Δ(G) ≥ 9, then χ'ₗ(G) = Δ(G) and χ''ₗ(G) = Δ(G)+1. We also show that if Δ(G) ≥ 6, then χ'ₗ(G) ≤ Δ(G)+1 and if Δ(G) ≥ 7, then χ''ₗ(G) ≤ Δ(G)+2. All of these results extend to graphs in the projective plane and when Δ(G) ≥ 7 the results also extend to graphs in the torus and Klein bottle. This second edge-choosability result improves on work of Wang and Lih and of Zhang and Wu. All of our results use the discharging method to prove structural lemmas about the existence of subgraphs with small degree-sum. For example, we prove that if G is a planar graph with no two 3-cycles sharing an edge and with Δ(G) ≥ 7, then G has an edge uv with d(u) ≤ 4 and d(u)+d(v) ≤ Δ(G)+2. All of our proofs yield linear-time algorithms that produce the desired colorings.
LA - eng
KW - list coloring; edge coloring; total coloring; Vizing's Conjecture; Vizing's conjecture
UR - http://eudml.org/doc/270180
ER -

References

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