Chromaticity of Bipartite Graphs With Seven Edges Deleted

Abstract

For integers \(p\), \(q\), \(s\) with \(p \geq q \geq 2\) and \(s \geq 0\), let \(\mathcal{K}_{2}^{-s}(p,q)\) denote the set of \(2\)-connected bipartite graphs which can be obtained from the complete bipartite graph \(K_{p,q}\) by deleting a set of \(s\) edges. F.M.Dong et al. (Discrete Math. vol.\(224 (2000) 107-124\)) proved that for any graph \(G \in \mathcal{K}_{2}^{-s}(p,q)\) with \(p \geq q \geq 3\) and \(0 \leq s \leq \min\{4, q-1\}\), then \(G\) is chromatically unique. In \([13]\), we extended this result to \(s = 5\) and \(s = 6\). In this paper, we consider the case when \(s = 7\).