On the Ohba Number of K_{3,b}: Exact Values for b ≤ 8 by a Certified Exhaustive Search

For integers a, b ≥ 2 let φ(a,b) be the least t ≥ 0 such that the join K_{a,b} + K_t of the complete bipartite graph K_{a,b} with the complete graph K_t is chromatic-choosable, that is, (t+2)-choosable. This is the Ohba number τ_0(a,b) of K_{a,b}; to determine it is a question of the Open Problem Garden, attributed there to J. Allagan and posted by J. Noel. The case a = 2 was settled by Cano, Gutknecht, Kappaganthula, Miller, Mudrock and Thornburgh, who also showed by a construction that K_{3,b} + K_t is not (t+2)-choosable when t ≥ 1 and b ≥ 1 + (t+2)⌈(t+2)/2⌉; a theorem of J. Zhu and X. Zhu implies that K_{3,b} + K_t is (t+2)-choosable when t ≥ 1 and b ≤ t+3. Let N_3(t) be the least b for which K_{3,b} + K_t is not (t+2)-choosable. We prove, by a computer-assisted exhaustive search with exact certificates, that K_{3,6} + K_1 is 3-choosable and that K_{3,8} + K_2 is 4-choosable. With the construction this gives N_3(1) = 7 and N_3(2) = 9, hence φ(3,b) = φ(b,3) = 1 for 3 ≤ b ≤ 6, φ(3,b) = φ(b,3) = 2 for b ∈ {7, 8}, and φ(3,b) ≥ 3 for b ≥ 9. The search reduces the question to a covering problem for the lists of the t+3 vertices outside the part of size b, runs, up to renaming of colours, through all these lists in which no colour is common to the three vertices of the part of size 3 (the others are settled at once; 551 cases for t = 1, 297,955 for t = 2), and settles every case either by counting configurations or by one of 18, respectively 435, dual solutions of a linear program, which are verified in exact rational arithmetic; the certificates and a program that checks them in less than a minute are provided. No novelty is claimed for the values φ(3,b), 5 ≤ b ≤ 8: they were not found in the sources read (the problem page, the two papers named above, a 2026 survey and two zbMATH entries), but Allagan's 2009 thesis, which according to its abstract treats the Ohba numbers of the graphs K(m,n,1,…,1) = K_{m,n} + K_t, could not be consulted and may contain some of them; neither could two 2015 papers of Allagan with coauthors and a 2007 paper of G. L. Chia and V. L. Chia. We conjecture that N_3(t) = 1 + (t+2)⌈(t+2)/2⌉ for all t; for t = 3 this is supported only by searches without certificates. The general question remains open. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OPG-59927 (Open Problem Garden, List Colourings of Complete Multipartite Graphs with 2 Big Parts, posted by J. Noel, 2014).