Ore's conjecture on color-critical graphs is almost true

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Abstract

A graph G is k-critical if it has chromatic number k, but every proper subgraph of G is (k1)-colorable. Let fk(n) denote the minimum number of edges in an n-vertex k-critical graph. We give a lower bound, fk(n)F(k,n), that is sharp for every n=1(modk1). The bound is also sharp for k=4 and every n6. 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 k4 and nk+2, fk(n+k1)=fk(n)+k12(k2k1) holds for each k4 for all but at most k3/12 values of n. We give a polynomial-time algorithm for (k1)-coloring of a graph G that satisfies |E(G[W])|<F(k,|W|) for all WV(G), |W|k. We also present some applications of the result.

Keywords

Graph coloring
k-critical graphs
Sparse graphs

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1

Research of this author is supported in part by NSF grants DMS-0965587 and DMS-1266016 and by grants 12-01-00448 and 12-01-00631 of the Russian Foundation for Basic Research.

2

Research of this author is partially supported by the Arnold O. Beckman Research Award of the University of Illinois at Urbana–Champaign and from National Science Foundation grant DMS 08-38434 “EMSW21-MCTP: Research Experience for Graduate Students.”