On the Attainability of Upper Bounds for the Circular Chromatic Number of <em>K</em>₄-Minor-Free Graphs.

• Main Author: Holt, Tracy Lance
• Format: Others
• Published: Digital Commons @ East Tennessee State University 2008
• Subjects: Graph Homomorphism; Circular Chromaitc Number; Circular Graphs; Graph Theory; Discrete Mathematics and Combinatorics; Mathematics; Physical Sciences and Mathematics
• Online Access: https://dc.etsu.edu/etd/1916; https://dc.etsu.edu/cgi/viewcontent.cgi?article=3268&amp;context=etd

Let G be a graph. For k ≥ d ≥ 1, a k/d -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 χc(G), is equal to the smallest k/d where a k/d -coloring exists. In [6], Pan and Zhu have given a function μ(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 μ(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 χc(Gg) = μ(g) if and only if k is not divisible by 3. In other words, for any odd g, the question of attainability of μ(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.