An edge coloring of hypergraph H is a function？？such that？？holds for any pair of intersecting edges .The minimum number of colors in edge
colorings of H is called the chromatic
index of H and is denoted by . Erdös, Faber and Lovász proposed a famous
conjecture that？？holds for any
loopless linear hypergraph H with n vertices. In this paper, we show
that？？is true for gap-restricted hypergraphs. Our result extends a result of Alesandroni in
2021.
References
[1]
Erdos, P. (1981) On the Combinatorial Problems Which I Would Most Like to See Solved. Combinatorica, 1, 25-42. https://doi.org/10.1007/BF02579174
[2]
Chang, W.I. and Lawler, E.L. (1988) Edge Coloring of Hypergraphs and a Conjecture of Erdos, Faber, Lovász. Combinatorica, 8, 293-295.
https://doi.org/10.1007/BF02126801
[3]
Seymour, P. (1982) Packing Nearly-Disjoint Sets. Combinatorica, 2, 91-97.
https://doi.org/10.1007/BF02579285
[4]
Kahn, J. and Seymour, P.D. (1992) A Fractional Version of the Erdos-Faber-Lovász Conjecture. Combinatorica, 12, 155-160. https://doi.org/10.1007/BF01204719
[5]
Kahn, J. (1992) Coloring Nearly-Disjoint Hypergraphs with n + o(n) Colors. Journal of Combinatorial Theory, Series A, 59, 31-39.
https://doi.org/10.1016/0097-3165(92)90096-D
[6]
Sánchez-Arroyo, A. (2008) The Erdos-Faber-Lovász Conjecture for Dense Hypergraphs. Discrete Mathematics, 308, 991-992.
https://doi.org/10.1016/j.disc.2007.09.026
[7]
Romero, D. and Alonso-Pecina, F. (2014) The Erdos-Faber-Lovász Conjecture Is True for n ≤ 12. Discrete Mathematics, Algorithms and Applications, 6, Article ID: 1450039. https://doi.org/10.1142/S1793830914500396
[8]
Faber, V. and Harris, D. (2019) Edge-Coloring Linear Hypergraphs with Medium- Sized Edges. Random Structures & Algorithms, 55, 153-159.
https://doi.org/10.1002/rsa.20843
[9]
Bretto, A., Faisant, A. and Hennecart, F., (2020) On Hyperedge Coloring of Weakly Trianguled Hypergraphs and Well Ordered Hypergraphs. Discrete Mathematics, 343. https://doi.org/10.1016/j.disc.2020.112059
[10]
Alesandroni, G. (2021) The Erdos-Faber-Lovász Conjecture for Weakly Dense Hypergraphs. Discrete Mathematics, 344.
https://doi.org/10.1016/j.disc.2021.112401
[11]
Kang, D., Kelly, T., Kühn, D., Methuku, A. and Osthus, D. (2023) A Proof of the Erdos-Faber-Lovász Conjecture. Annals of Mathematics, 198, 537-618.
https://doi.org/10.4007/annals.2023.198.2.2
[12]
Wang, Q. and Zhang, X. (2023) A Note on Edge Coloring of Linear Hypergraphs. Journal of Mathematical Research with Applications, 43, 535-541.
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