On the Attainability of Upper Bounds for the Circular Chromatic Number of K4-Minor-Free Graphs.

Holt, Tracy Lance

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East Tennessee State University

On the Attainability of Upper Bounds for the Circular Chromatic Number of K4-Minor-Free Graphs.

dc:creator Holt, Tracy Lance

thesis:degree_name MS (Master of Science)

dc:date.issued2008

Abstract

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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.

Degree

thesis:*

Name thesis:degree_name: MS (Master of Science)
Level thesis:degree_level: Thesis - unrestricted
Discipline thesis:degree_discipline: Mathematical Sciences
Year dc:date.issued: 2008

Author and committee

dc:creator, dc:contributor.*

Author dc:creator: Holt, Tracy Lance

Subjects

dc:subject × 7

Rights

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Statement dc:rights: Copyright by the authors.

Identifiers

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Repository record dc:identifier: https://dc.etsu.edu/etd/1916
OAI identifier oai:identifier: oai:dc.etsu.edu:etd-3268

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