Abstract
A graph G is k-critical if it has chromatic number k, but every proper subgraph of G is (kâ1)-colorable. Let fk(n) denote the minimum number of edges in an n-vertex k-critical graph. We give a lower bound, fk(n)â„F(k,n), that is sharp for every n=1(modkâ1). The bound is also sharp for k=4 and every nâ„6. The result improves a bound by Gallai and subsequent bounds by Krivelevich and Kostochka and Stiebitz, and settles the corresponding conjecture by Gallai from 1963. It establishes the asymptotics of fk(n) for every fixed k. It also proves that the conjecture by Ore from 1967 that for every kâ„4 and nâ„k+2, fk(n+kâ1)=fk(n)+kâ12(kâ2kâ1) holds for each kâ„4 for all but at most k3/12 values of n. We give a polynomial-time algorithm for (kâ1)-coloring of a graph G that satisfies |E(G[W])|<F(k,|W|) for all WâV(G), |W|â„k. We also present some applications of the result.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have