On the chromatic index of multigraphs without large triangles

Research output: Contribution to journalArticle

32 Citations (Scopus)

Abstract

Let M be a multigraph. Vizing (Kibernetika (Kiev) 1 (1965), 29-39) proved that χ′(M)≤Δ(M)+μ(M). Here it is proved that if χ′(M)≥Δ(M)+s, where 1 2(μ(M) + 1) < s then M contains a 2s-sided triangle. In particular, (C′) if μ(M)≤2 and M does not contain a 4-sided triangle then χ′(M)≤Δ(M) + 1. Javedekar (J. Graph Theory 4 (1980), 265-268) had conjectured that (C) if G is a simple graph that does not induce K1,3 or K5-e then χ(G)≤ω(G) + 1. The author and Schmerl (Discrete Math. 45 (1983), 277-285) proved that (C′) implies (C); thus Javedekar's conjecture is true.

Original languageEnglish (US)
Pages (from-to)156-160
Number of pages5
JournalJournal of Combinatorial Theory, Series B
Volume36
Issue number2
DOIs
StatePublished - 1984
Externally publishedYes

Fingerprint

Chromatic Index
Multigraph
Graph theory
Triangle
Simple Graph
Imply

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

On the chromatic index of multigraphs without large triangles. / Kierstead, Henry.

In: Journal of Combinatorial Theory, Series B, Vol. 36, No. 2, 1984, p. 156-160.

Research output: Contribution to journalArticle

@article{f36b2eef2b4147fc907721ce3aee23e0,
title = "On the chromatic index of multigraphs without large triangles",
abstract = "Let M be a multigraph. Vizing (Kibernetika (Kiev) 1 (1965), 29-39) proved that χ′(M)≤Δ(M)+μ(M). Here it is proved that if χ′(M)≥Δ(M)+s, where 1 2(μ(M) + 1) < s then M contains a 2s-sided triangle. In particular, (C′) if μ(M)≤2 and M does not contain a 4-sided triangle then χ′(M)≤Δ(M) + 1. Javedekar (J. Graph Theory 4 (1980), 265-268) had conjectured that (C) if G is a simple graph that does not induce K1,3 or K5-e then χ(G)≤ω(G) + 1. The author and Schmerl (Discrete Math. 45 (1983), 277-285) proved that (C′) implies (C); thus Javedekar's conjecture is true.",
author = "Henry Kierstead",
year = "1984",
doi = "10.1016/0095-8956(84)90022-4",
language = "English (US)",
volume = "36",
pages = "156--160",
journal = "Journal of Combinatorial Theory. Series B",
issn = "0095-8956",
publisher = "Academic Press Inc.",
number = "2",

}

TY - JOUR

T1 - On the chromatic index of multigraphs without large triangles

AU - Kierstead, Henry

PY - 1984

Y1 - 1984

N2 - Let M be a multigraph. Vizing (Kibernetika (Kiev) 1 (1965), 29-39) proved that χ′(M)≤Δ(M)+μ(M). Here it is proved that if χ′(M)≥Δ(M)+s, where 1 2(μ(M) + 1) < s then M contains a 2s-sided triangle. In particular, (C′) if μ(M)≤2 and M does not contain a 4-sided triangle then χ′(M)≤Δ(M) + 1. Javedekar (J. Graph Theory 4 (1980), 265-268) had conjectured that (C) if G is a simple graph that does not induce K1,3 or K5-e then χ(G)≤ω(G) + 1. The author and Schmerl (Discrete Math. 45 (1983), 277-285) proved that (C′) implies (C); thus Javedekar's conjecture is true.

AB - Let M be a multigraph. Vizing (Kibernetika (Kiev) 1 (1965), 29-39) proved that χ′(M)≤Δ(M)+μ(M). Here it is proved that if χ′(M)≥Δ(M)+s, where 1 2(μ(M) + 1) < s then M contains a 2s-sided triangle. In particular, (C′) if μ(M)≤2 and M does not contain a 4-sided triangle then χ′(M)≤Δ(M) + 1. Javedekar (J. Graph Theory 4 (1980), 265-268) had conjectured that (C) if G is a simple graph that does not induce K1,3 or K5-e then χ(G)≤ω(G) + 1. The author and Schmerl (Discrete Math. 45 (1983), 277-285) proved that (C′) implies (C); thus Javedekar's conjecture is true.

UR - http://www.scopus.com/inward/record.url?scp=0012315875&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0012315875&partnerID=8YFLogxK

U2 - 10.1016/0095-8956(84)90022-4

DO - 10.1016/0095-8956(84)90022-4

M3 - Article

AN - SCOPUS:0012315875

VL - 36

SP - 156

EP - 160

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 2

ER -