Chromatically unique bipartite graphs with low 3-independent partition numbers Original Research Article

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 at most 2p−1+2q−1+s+2, then G is χ-unique. It follows that any graph in K2−s(p,q) is χ-unique if p⩾q⩾3 and 1⩽s⩽min{q−1,4}.