Adding a vertex s adjacent to every vertex of K₃,₄ (whose two parts are already fully joined to each other) turns G into the complete tripartite graph K₁,₃,₄, with parts {s}, X (size 3), Y (size 4). Degrees: each vertex of X has degree 4 (to Y) + 1 (to s) = 5; each vertex of Y has degree 3 (to X) + 1 (to s) = 4; s has degree 3 + 4 = 7. So the maximum degree Δ = 7 (at s). By Vizing's theorem the edge-chromatic number is either Δ or Δ+1, and complete multipartite graphs are known to be 'Class 1' (edge-chromatic number exactly Δ) except for the single special case of an odd complete graph — which G is not here. So the minimum number of colours is Δ = 7.