The chromatic sum of a graph is the minimum total of the colors on the vertices taken over all possible proper colorings using positive integers. Erdös et al [Graphs that require many colors to achieve their chromatic sum, Congr. Numer. 71 (1990) 17–28.] considered the question of finding graphs with minimum number of vertices that require t colors beyond their chromatic number k to obtain their chromatic sum. The number of vertices of such graphs is denoted by P ( k , t ) . They presented some upper bounds for this parameter by introducing certain constructions. In this paper we give some lower bounds for P ( k , t ) and considerably improve the upper bounds by introducing a class of graphs, called tabular graphs. Finally, for fixed t and sufficiently large k the exact value of P ( k , t ) is determined.