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

It is proved that if GG is a graph containing a spanning tree with at most three leaves, then the chromatic polynomial of GG has no roots in the interval (1,t1](1,t1], where t1≈1.2904t1≈1.2904 is the smallest real root of the polynomial (t−2)6+4(t−1)2(t−2)3−(t−1)4(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 t1t1 from above. We employ the Whitney 2-switch operation to manage the analysis of an infinite class of chromatic polynomials.