Completing Partial Latin Squares with One Nonempty Row, Column, and Symbol

  • Jaromy Kuhl
  • Michael W. Schroeder
Keywords: Partial latin square, Completing


Let $r,c,s\in\{1,2,\ldots,n\}$ and let $P$ be a partial latin square of order $n$ in which each nonempty cell lies in row $r$, column $c$, or contains symbol $s$. We show that if $n\notin\{3,4,5\}$ and row $r$, column $c$, and symbol $s$ can be completed in $P$, then a completion of $P$ exists. As a consequence, this proves a conjecture made by Casselgren and Häggkvist. Furthermore, we show exactly when row $r$, column $c$, and symbol $s$ can be completed.

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