Claims

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1992_10_01_andersen: Andersen's Theorem 1 (Discrete Math. 1992) gives every graph of maximum degree at most three a strong edge-coloring with at most ten colors, so the statement holds for maximum degree at most three; refereed; partial.

1993_06_01_horak_he_trotter: The Theorem of Horák, He and Trotter (J. Graph Theory 1993): a graph of maximum degree at most three has strong chromatic index at most ten, best possible; the statement holds for maximum degree at most three; refereed.

2000_10_01_mahdian: Mahdian's theorem (Toronto M.Sc. thesis; Random Structures Algorithms 17 (2000)): a C_4-free graph of large maximum degree has strong chromatic index at most (2 + o(1)) Delta^2 / log Delta; refereed.

2002_01_01_vu: Vu's list-coloring theorem for locally sparse graphs (Combin. Probab. Comput. 2002) extends Mahdian's bound: graphs without a fixed bipartite H have strong chromatic index O_H(Delta^2 / log Delta); claimed, partial.

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