Structures and chromaticity of some extremal 3-colourable graphs

Given a graph G and a positive integer r, let sr(G)=P(G; r)=r!. Thus (G)=r and sr(G)=1 i G is uniquely r-colourable. It is known that if G is uniquely 3-colourable, then e(G)¿2v(G)−3. In this paper, we show that if G is a 3-colourable connected graph with e(G)=2v(G)−k where k¿4, then s3(G)¿2k−3; and if, further, G is 2-connected and s3(G)=2k−3, then t(G)6v(G)−k where t(G) denotes the number of C3’s in G. We proceed to determine the structures of all 3-colourable 2-connected graphs G with e(G) = 2v(G)− k; s3(G) = 2k−3 and t(G) = v(G)− k. By applying this structural result, we nally study the chromaticity of such graphs and produce new chromatically equivalent classes. c © 1999 Elsevier Science B.V. All rights reserved.