What is the list-chromatic number of Sudoku

Definitions and mathematical statement

Let $G_{\mathrm{Sud}}$ have vertex set $\{1,\ldots,9\}^2$, one vertex for each cell of the standard Sudoku board. Distinct cells $(r,c),(r',c')$ are adjacent when $r=r'$, or $c=c'$, or \[ \left\lfloor\frac{r-1}{3}\right\rfloor=\left\lfloor\frac{r'-1}{3}\right\rfloor, \qquad \left\lfloor\frac{c-1}{3}\right\rfloor=\left\lfloor\frac{c'-1}{3}\right\rfloor, \] so they lie in one $3\times3$ block. A list assignment gives each cell $v$ a finite set $L(v)$ of available symbols; the symbol sets need not coincide. A proper list coloring chooses $c(v)\in L(v)$ with $c(v)\ne c(w)$ for every adjacent pair.

Problem. Determine \[ \chi_{\ell}(G_{\mathrm{Sud}})=\min\{k\geq1:\text{every assignment with }|L(v)|=k \text{ admits a proper list coloring}\}. \] There are no prefilled cells or other constraints beyond the stated rows, columns, and blocks. The list size is uniform, but the contents are arbitrary. Each row forms a nine-vertex clique, so any interpretation involving list size $4$ cannot be a positive universal-colorability claim for this same board.

The source nevertheless says even for $k=4$ the question is open. That sentence does not consistently identify a subcase of the defined nine-by-nine problem; it might refer to another board-size parameter, but no such variant is specified. The research summary speculates that the answer may be $9$ by an analogue of Galvin's theorem while explicitly admitting that it was not verified. That speculation is not accepted as a proved list-coloring result. The source’s Dinitz/Galvin comparison is an analogy, not an established reduction.

Short English statement

Find the list-chromatic number of the standard Sudoku graph with all row, column, and block conflicts. The source’s $k=4$ sentence and its speculative claim of value $9$ are unresolved source issues.

Sources

• Cooper, Combinatorial Problems I Like, Graphs, Hypergraphs, Set Systems, Designs, source-order bullet 24, October 2020 snapshot, https://people.math.sc.edu/cooper/combprob.html; HTML extraction, accessed 2026-07-29. Later resolutions were not checked item by item according to the record.
• Dinitz conjecture / Galvin theorem, named in the statement as an analogue. No specific Sudoku list-coloring theorem, Galvin reduction, or bibliographic identifier is supplied.
• Local registry data/unsolvedmath/problems.json, permanent ID $3100024$, source code catalogue entry, What is the list-chromatic number of Sudoku. The record labels this open and OPEN-TRIAGE, with NEEDS_REVIEW source-status and rights qualifications.