Critical sets for Sudoku and general graph colorings

Abstract

We discuss the problem of finding critical sets in graphs, a concept which has appeared in a number of guises in the combinatorics and graph theory literature. The case of the Sudoku graph receives particular attention, because critical sets correspond to minimal fair puzzles. We define four parameters associated with the sizes of extremal critical sets and (a) prove several general results about these parameters' properties, including their computational intractability, (b) compute their values exactly for some classes of graphs, (c) obtain bounds for generalized Sudoku graphs, and (d) offer a number of open questions regarding critical sets and the aforementioned parameters.

Extremal and monotone behaviour of the Sudoku number and related critical set parameters

“…One would expect that sn(K n K n , k) is non-decreasing in terms of k. In Section 4, we observe that scs(G, k) and lcs(G, k) are monotone non-decreasing in k for every graph G, while this is not the case for sn(G, k) and lcs(G, k). This answers a question of [7]. We also consider this question about the monotone behaviour of the parameters for the subgraph-relation, under the (necessary) condition of fixed chromatic number.…”

“…The Sudoku number has been defined before as smallest defining set (see e.g. [14] and references therein), smallest critical set [7] or forcing chromatic number [10]. A similar notion is the forced matching number of graphs, see [1].…”

“…We will prefer to use sn(G) over scs(G), as this highlights that it is the most natural and most studied parameter of the four numbers associated with critical sets. Considering k-colourings with k > χ(G) has been done in [16] and was suggested in Cooper and Kirkpatrick [7] as well, but most research done so far was for the case k = χ. In particular, the Sudoku number of certain graphs have been determined in the past.…”

Sudoku Rectangle Completion (Extended Abstract)

“…Concepts of "critical (or defining) sets" i.e. "the minimum Sudoku problem" and "Sudoku trades (or detection of unavoidable sets in Sudoku)" are studied in [9], [10], [14], and [3]. In [1] Sudoku is considered as a special case of Gerechte designs and they introduce some interesting mathematical problems about them.…”

Spectrum of free-form Sudoku graphs

“…For example, there has been some research interest in counting the number of possible completions of a partially colored Sudoku graph. A related question is how to (minimally) partially color the graph such that there is only one way to complete the coloring to a valid Sudoku solution [4]. It has even been shown that 16 precolored vertices do not suffice in order to guarantee a unique completion, however by means of computational brute force [20].…”