A Fixed-Part Bipartite Extremal Bound for the Eight-Vertex Spider

Abstract

Let T be the eight-vertex tree obtained from a seven-vertex path by adjoining a leaf to its central vertex, equivalently the spider with leg lengths 3, 3, and 1. Waite and Aydin (arXiv:2607.29579) identify T as the smallest tree outside the scope of their Theorem 1.4. We prove the bound predicted by their fixed-part conjecture: every bipartite graph G of order N with more than 2N edges contains T. Consequently, ex_bip(m,n;T) ≤ 2(m+n) for every pair of part-sizes m,n. Combined with the Waite–Aydin theorem, this establishes both forms of their conjecture for every tree on at most nine vertices. The proof extracts a subgraph of minimum degree at least 3, finds a vertex of degree at least 5 in a smallest bipartition class, and applies a tailored rooted embedding lemma. The upper bound is attained for infinitely many balanced part-sizes; for general m,n ≥ 2 it is sharp within an additive constant of 8. Draft circulated for independent expert review. Generative-AI tools assisted with literature triage, proof exploration, and manuscript preparation; the argument is included in full so that every inference can be checked directly. The ancillary script census.py reproduces the tree enumeration reported in Remark 7.3.