Lecture notes on sparse color-critical graphs

This text together with the attached paper [8] surveys results on color-critical graphs, with emphasis on sparse ones. The first two sections discuss the important contributions by Dirac and Gallai and present proofs of some remarkable results of them. The next two sections discuss the later progress and a number of applications of the recent results. We also use [8] for description of some applications. In Section 6 we present a proof for 4-critical graphs of a conjecture of Gallai on sparsest color-critical graphs. In the last section, we briefly survey similar problems for hypergraphs and triangle-free graphs and mention some unsolved problems. Recall that a (proper) k-coloring of a graph G is a mapping g : V (G) → {1, . . . , k} such that g(v) 6= g(u) for each vu ∈ E(G). The minimum k such that G has a k-coloring is the chromatic number of G, denoted by χ(G). For a positive integer k, a graph G is k-critical if χ(G) = k, but every proper subgraph of G is (k − 1)-colorable. It is easy to check that the complete k-vertex graph Kk is k-critical and that each odd cycle is 3-critical.