The Bichromatic Triangle Coloring Polynomial of Some 2-Trees

The bichromatic triangle polynomial PG(k) counts vertex k-colorings in which every triangle uses exactly two colors. We develop a transfer matrix framework for three canonical families of 2-trees: book graphs Bn, 1-fans Fn1, and triangulated ladders TLm. In each case, PG(k) satisfies a second-order linear recurrence with an explicit closed form; for TLm this yields a Chebyshev representation, while for Fn1 the binary specialization gives PFn1(2)=2Fn+1. A spectral identity α2=r+ links the dominant characteristic roots of the fan and ladder recurrences, implying identical exponential growth rates when indexed by vertex count, whereas book graphs grow strictly faster for k≥4. In fact, this correspondence is exact: for all k≥2, the triangulated ladder polynomial coincides with that of a suitably indexed 1-fan. Passing to line graphs, we interpret PL(Kn)(k) as counting edge colorings of Kn that forbid both monochromatic and rainbow triangles, and we identify a sharp obstruction threshold at n≥6.