Chromatically Unique Bipartite Graphs with Certain 3-independent Partition Numbers III

For integers p, q, s with p ≥ q ≥ 2 and s ≥ 0 , let K2−s (p,q) denote the set of 2_connected bipartite graphs which can be obtained from K(p,q) by deleting a set of s edges. In this paper, we prove that for any graph GЄK2 − s p,q with p ≥ q ≥ 3 and 1 ≤ s ≤ q - 1 if the number of 3-independent partitions of G is 2p-1 + 2q-1 + s + 4, then G is chromatically unique. This result extends both a theorem by Dong et al. [2]; and results in [4] and [5]