| Let G be a graph. For k ≥ d ≥ 1, a kd -coloring of G is a coloring c of vertices of G with colors 0, 1, 2,..., k - 1, such that d ≤ |c( x) - c(y)| ≤ k - d, whenever xy is an edge of G. We say that the circular chromatic number of G, denoted chic(G), is equal to the smallest kd where a kd -coloring exists. In [6], Pan and Zhu have given a function mu( g) that gives an upper bound for the circular-chromatic number for every K4-minor-free graph Gg of odd girth at least g, g ≥ 3. In [7], they have shown that their upper bound in [6] can not be improved by constructing a sequence of graphs approaching mu(g) asymptotically. We prove that for every odd integer g = 2k + 1, there exists a graph Gg ∈ G /K4 of odd girth g such that chic(Gg) = mu( g) if and only if k is not divisible by 3. In other words, for any odd g, the question of attainability of mu( g) is answered for all g by our results. Furthermore, the proofs [6] and [7] are long and tedious. We give simpler proofs for both of their results. |
