Problems in generalized graph colorings
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) ≤ eOlogn . We go on to show that f(n, 6, 5) ≤ eOlogn and f(n, 7, 6) ≤ eOlogn , 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 &ceill0; log2 p &ceilr0; − 4 + 4p2log 2p , then f(n, p, q) ≤ eOlogn . This improves upon a result of Eichhorn and Mubayi. ^
Zachary A Kudlak,
"Problems in generalized graph colorings"
Dissertations and Master's Theses (Campus Access).