Problems in generalized graph colorings

Author

Zachary A. Kudlak, University of Rhode Island

Date of Award

2010

Degree Type

Dissertation

Abstract

We study a problem related to generalized graph Ramsey numbers. An edge coloring is said to be a (p, q)-edge coloring of the graph Kn if on each induced subgraph on p vertices there are at least q colors realized on its edges. The function f(n, p, q) is the least number of colors necessary for which a (p, q)-edge coloring of Kn exists. Erdős and Gyárfás posed the following question concerning a lower bound on f(n, p, p − 1). If p ≥ 3, is f(n, p, p − 1) ≥ cn&epsis;, for positive c and &epsis; that only depend on p? We explore the structural properties of two explicit edge colorings. Utilizing these properties we give an independent proof that f( n, 5, 4) ≤ [special characters omitted]. We go on to show that f(n, 6, 5) ≤ [special characters omitted] and f(n, 7, 6) ≤ [special characters omitted], which were previously unknown results, and in the process give a partial answer to the question asked by Erdős and Gyárfás. Additionally, we show that if p ≥ 6 and q = 2[special characters omitted]log2 p[special characters omitted] − 4 + [special characters omitted], then f(n, p, q) ≤ [special characters omitted]. This improves upon a result of Eichhorn and Mubayi.

Recommended Citation

Kudlak, Zachary A., "Problems in generalized graph colorings" (2010). Open Access Dissertations. Paper 2349.
https://digitalcommons.uri.edu/oa_diss/2349