A Characterization of the Doob Graphs

For a graph Γ we say that λ exists if every edge lies in exactly λ triangles. Egawa has shown that the Hamming graphs are determined by their parameters, unless λ equals 2. In that case the only possible graphs are the Doob graphs. Rifà and Huguet and Nomura generalised this result by Egawa for λ ≠ 2. We generalise Egawa′s result for λ = 2. The main part of the proof is a characterization of cartesian products of graphs.