A graph G is chromatically unique if its chromatic polynomial completely determines the graph. An n-spoked wheel, Wn, is shown to be chromatically unique when n ≥ 4 is even [S.-J. Xu and N.-Z. Li, The chromaticity of wheels, Discrete Math. 51 (1984) 207–212]. When n is odd, this problem is still open for n ≥ 15 since 1984, although it was shown by di erent researchers that the answer is no for n = 5, 7, yes for n = 3, 9, 11, 13, and unknown for other odd n. We use the beta invariant of matroids to prove that if M is a 3-connected matroid such that |E(M)| = |E(Wn)| and β (M) = β (M(Wn)), where β (M) is the beta invariant of M, then M ≅ M(Wn). As a consequence, if G is a 3-connected graph such that the chromatic (or flow) polynomial of G equals to the chromatic (or flow) polynomial of a wheel, then G is isomorphic to the wheel. The examples for n = 3, 5 show that the 3-connectedness condition may not be dropped. We also give a splitting formula for computing the beta invariants of general parallel connection of two matroids as well as the 3-sum of two binary matroids. This generalizes the corresponding result of Brylawski [A combinatorial model for series-parallel networks, Trans. Amer. Math. Soc. 154 (1971) 1–22].
Lee, S., & Wu, H. (2021). Beta invariant and chromatic uniqueness of wheels. Discussiones Mathematicae Graph Theory. https://doi.org/10.7151/dmgt.2444