Non-chordal graphs having integral-root chromatic polynomials II

It is known that the chromatic polynomial of any chordal graph has only integer roots. However, there also exist non-chordal graphs whose chromatic polynomials have only integer roots. Dmitriev asked in 1980 if for any integer p⩾4, there exists a graph with chordless cycles of length p whose chromatic polynomial has only integer roots. This question has been given positive answers by Dong and Koh for p=4 and p=5. In this paper, we construct a family of graphs in which all chordless cycles are of length p for any integer p⩾4. It is shown that the chromatic polynomial of such a graph has only integer roots iff a polynomial of degree p−1 has only integer roots. By this result, this paper extends Dong and Kohʹs result for p=5 and answer the question affirmatively for p=6 and 7.