Chromatically Unique Bipartite Graphs With Certain 3-independent Partition Numbers

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 Kp,q by deleting a set of s edges. In this paper, we prove that for any graph G K2-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 + 3, then G is chromatically unique. This result extends the similar theorem by Dong et al. (Discrete Math. vol. 224 (2000) 107-124).