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

Discussiones Mathematicae Graph Theory (2009)

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

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topDaniel 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 -

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