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Defining sets in vertex colorings of graphs and latin rectangles. (English) Zbl 0879.05028
A set $$S$$ of vertices of a graph $$G$$ is called a defining set of vertex colourings of $$G$$ if there exists a proper colouring $$\varphi$$ of $$S$$ using $$k\leq \chi(G)$$ colours such that $$\varphi$$ has a unique extension to a $$\chi (G)$$-colouring of $$G$$. The cardinality of a minimum defining set of $$G$$ is denoted by $$d_v (G)$$. A critical set in an $$m \times n$$ array, $$m\leq n$$, is a set $$S$$ of cells which are preoccupied with $$k\leq n$$ symbols such that there is a unique extension of $$S$$ to a latin rectangle of size $$m\times n$$.
Some lower bounds for $$d_v (G)$$ are obtained and used to show that \begin{aligned} d_v & (K_2 \times C_{2n+1}) = n+1; \\ d_v & (C_m \times K_n)= m(n-3) \text{ for }n \geq 6; \\ d_v & (P_m \times K_n) =m(n-3) +2 \text{ for } n\geq 6; \text{ and} \\ d_v & (K_m \times K_n) =m(n-m) \text{ if }n \geq m^2, \end{aligned} where $$P_m \times K_n$$ is the Cartesian product of $$K_2$$ and $$C_{2n+1}$$.
The authors also determine the size of smallest critical sets of a back circulant latin rectangle of size $$m\times n$$, with $$2m\leq n$$. It is conjectured that for any latin square of order $$n$$, the size of a critical set is at least $$\lfloor {n^2 \over 4} \rfloor$$.
Reviewer’s remark: Thayer Morrill and Dan Pritikin have studied defining sets and list-defining sets $$S$$ in graphs in a slightly broader sense that a proper $$k$$-colouring $$\varphi$$ of $$S$$ has a unique extension to a proper $$k$$-colouring of $$G$$, where $$\Delta (G)\leq k\leq \Delta (G)+1$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05B15 Orthogonal arrays, Latin squares, Room squares
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