A zero-free interval for chromatic polynomials of graphs with 3-leaf spanning trees

Abstract

a b s t r a c tIt is proved that if G is a graph containing a spanning tree with at most three leaves, then the chromatic polynomial of G has no roots in the interval (1, t 1 ], where t 1 ≈ 1.2904 is the smallest real root of the polynomial (t −2) 6 +4(t −1) 2 (t −2) 3 −(t −1) 4 . We also construct a family of graphs containing such spanning trees with chromatic roots converging to t 1 from above. We employ the Whitney 2-switch operation to manage the analysis of an infinite class of chromatic polynomials.

Chromatic Roots and Minor-Closed Families of Graphs

“…We remark that in [7], the present author proved an analogue of Thomassen's result for a slightly more general class of graphs. Theorem 3.1.…”

“…Now by the induction hypothesis of (b), we have Q(H + uv, t) ≥ βQ(H/uv, t). Substituting into (7) and using Lemma 4.2(iii) gives s ≥ [(1 − γ)(2 − t)β − γ(t − 1)]Q(H/uv, t) ≥ 0.…”