The chromaticity of complete bipartite graphs with at most one edge deleted

Let K(p, q), p ≤ q , denote the complete bipartite graph in which the two partite sets consist of p and q vertices, respectively. In this paper, we prove that (1) the graph K(p, q) is chromatically unique if p ≥ 2; and (2) the graph K(p, q) ‐ e obtained by deleting an edge e from K(p, q) is chromatically unique if p ≥ 3. The first result was conjectured by Salzberg, López, and Giudici, who also proved the second result under the condition that q ‐ p ≤ 1.