### Abstract

For a graph G = (V(G), E(G)), a strong edge coloring of G is an edge coloring in which every color class is an induced matching. The strong chromatic index of G, χ_{s} (G), is the smallest number of colors in a strong edge coloring of G. The strong chromatic index of the random graph G(n, p) was considered in Discrete Math. 281 (2004) 129, Austral. J. Combin. 10 (1994) 97, Austral. J. Combin. 18 (1998) 219 and Combin. Probab. Comput. 11 (1) (2002) 103. In this paper, we consider χ_{S}(G) for a related class of graphs G known as uniform or ε-regular graphs. In particular, we prove that for 0 < ε ≪ d < 1, all (d, ε)-regular bipartite graphs G = (U ∪ V, E) with |U| = |V|≥n_{0}(d, ε) satisfy χ _{s}(G)≤ξ(ε)δ(G)^{2}, where ξ(ε) → 0 as → 0 (this order of magnitude is easily seen to be best possible). Our main tool in proving this statement is a powerful packing result of Pippenger and Spencer (Combin. Theory Ser. A 51(1) (1989) 24).

Original language | English (US) |
---|---|

Pages (from-to) | 219-223 |

Number of pages | 5 |

Journal | Discrete Mathematics |

Volume | 286 |

Issue number | 3 |

DOIs | |

State | Published - Sep 28 2004 |

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### Keywords

- Strong chromatic index
- The regularity lemma

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*286*(3), 219-223. https://doi.org/10.1016/j.disc.2004.04.011

**Strong edge colorings of uniform graphs.** / Czygrinow, Andrzej; Nagle, Brendan.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 286, no. 3, pp. 219-223. https://doi.org/10.1016/j.disc.2004.04.011

}

TY - JOUR

T1 - Strong edge colorings of uniform graphs

AU - Czygrinow, Andrzej

AU - Nagle, Brendan

PY - 2004/9/28

Y1 - 2004/9/28

N2 - For a graph G = (V(G), E(G)), a strong edge coloring of G is an edge coloring in which every color class is an induced matching. The strong chromatic index of G, χs (G), is the smallest number of colors in a strong edge coloring of G. The strong chromatic index of the random graph G(n, p) was considered in Discrete Math. 281 (2004) 129, Austral. J. Combin. 10 (1994) 97, Austral. J. Combin. 18 (1998) 219 and Combin. Probab. Comput. 11 (1) (2002) 103. In this paper, we consider χS(G) for a related class of graphs G known as uniform or ε-regular graphs. In particular, we prove that for 0 < ε ≪ d < 1, all (d, ε)-regular bipartite graphs G = (U ∪ V, E) with |U| = |V|≥n0(d, ε) satisfy χ s(G)≤ξ(ε)δ(G)2, where ξ(ε) → 0 as → 0 (this order of magnitude is easily seen to be best possible). Our main tool in proving this statement is a powerful packing result of Pippenger and Spencer (Combin. Theory Ser. A 51(1) (1989) 24).

AB - For a graph G = (V(G), E(G)), a strong edge coloring of G is an edge coloring in which every color class is an induced matching. The strong chromatic index of G, χs (G), is the smallest number of colors in a strong edge coloring of G. The strong chromatic index of the random graph G(n, p) was considered in Discrete Math. 281 (2004) 129, Austral. J. Combin. 10 (1994) 97, Austral. J. Combin. 18 (1998) 219 and Combin. Probab. Comput. 11 (1) (2002) 103. In this paper, we consider χS(G) for a related class of graphs G known as uniform or ε-regular graphs. In particular, we prove that for 0 < ε ≪ d < 1, all (d, ε)-regular bipartite graphs G = (U ∪ V, E) with |U| = |V|≥n0(d, ε) satisfy χ s(G)≤ξ(ε)δ(G)2, where ξ(ε) → 0 as → 0 (this order of magnitude is easily seen to be best possible). Our main tool in proving this statement is a powerful packing result of Pippenger and Spencer (Combin. Theory Ser. A 51(1) (1989) 24).

KW - Strong chromatic index

KW - The regularity lemma

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UR - http://www.scopus.com/inward/citedby.url?scp=4444248242&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2004.04.011

DO - 10.1016/j.disc.2004.04.011

M3 - Article

AN - SCOPUS:4444248242

VL - 286

SP - 219

EP - 223

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 3

ER -