Let K(m,n)-2 denote the bipartite graph obtained by deleting two edges from the complete bipartite graph K(m,n) . In this paper, we prove that (1) K(m,n)-2 is chromatically unique if 3≤m≤n and n+m-((n-m)2+8)1/2>(n-m)2/2+4 . (2) K(m,m)-2, K(m,m+1)-2 and K(m,m+2)-2 are all chromatically unique if m≥3 . |