On Chromatic Uniqueness of a Family of K4-Homeomorphs II

Let P (G,λ) be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted G ∼ H, if P (G,λ) = P (H,λ). We write [G] = {H |H ∼ G}. If [G] = {G}, then G is said to be chromatically unique. A K4-homeomorph denoted by K4(a, b, c, d, e, f) if the six edges of complete graph K4 are replaced by the six paths of length a, b, c, d, e, f respectively. In this paper, we study the chromatically unique of such K4-homeomorph with girth 3a + 1, where b = a, c = a + 1 and d, e, f ≥ a.