On the minimal number of edges in color-critical graphs

Abstract

A graphG isk-critical if it has chromatic numberk, but every proper subgraph of it is (k−1)-colorable. This paper is devoted to investigating the following question: for givenk andn, what is the minimal number of edges in ak-critical graph onn vertices, with possibly some additional restrictions imposed? Our main result is that for everyk≥4 andn>k this number is at least $$\left( {\frac{{k - 1}}{2} + \frac{{k - 3}}{{2(k^2 - 2k - 1)}}} \right)n$$ , thus improving a result of Gallai from 1963. We discuss also the upper bounds on the minimal number of edges ink-critical graphs and provide some constructions of sparsek-critical graphs. A few applications of the results to Ramsey-type problems and problems about random graphs are described.