WOW 894 proof analysis: Petersen graph as the unique extremal case — girth-5 upper bound

The proof that blue independence ≤ 2·girth for cubic triangle-free graphs hinges on Petersen as the bottleneck. A set of 9 vertices mutually within distance 2 in a cubic triangle-free graph forces the graph to be isomorphic to Petersen (10 vertices, girth 5). Since Petersen has blue independence exactly 10 = 2·5, and the Wagner graph (girth 4) has blue independence 8 = 2·4, the bound is tight. For graphs with blue independence ≥ 11, a girth of at least 6 would be required — but no cubic triangle-free graph with both properties exists, making 894 a structural necessity theorem rather than a numerical coincidence.