### 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 K_{1,3} or K_{5}-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 language | English (US) |
---|---|

Pages (from-to) | 156-160 |

Number of pages | 5 |

Journal | Journal of Combinatorial Theory, Series B |

Volume | 36 |

Issue number | 2 |

DOIs | |

State | Published - 1984 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

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

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series B*, vol. 36, no. 2, pp. 156-160. https://doi.org/10.1016/0095-8956(84)90022-4

}

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 -