Defining sets in vertex colorings of graphs and latin rectangles Original Research Article

In a given graph G, a set of vertices S with an assignment of colors is said to be a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a χ(G)-coloring of the vertices of G. The concept of a defining set has been studied, to some extent, for block designs and also under another name, a critical set, for latin squares. In this note we extend this concept to graphs, and show its relationship with the critical sets of latin rectangles. The size of smallest defining sets for some classes of graphs are determined and a lower bound is introduced for an arbitrary graph G. The size of smallest critical sets of a back circulant latin rectangle of size m × n, with 2m ⩽ n, is also determined.