# Problems in generalized graph colorings

#### 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 *K _{n}* 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

*K*exists. Erdős and Gyárfás posed the following question concerning a lower bound on

_{n}*f*(

*n*,

*p*,

*p*− 1). If

*p*≥ 3, is

*f*(

*n*,

*p*,

*p*− 1) ≥

*cn*, for positive

^{&epsis;}*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) ≤ e

^{Ologn }. We go on to show that

*f*(

*n*, 6, 5) ≤ e

^{Ologn }and

*f*(

*n*, 7, 6) ≤ e

^{Ologn }, 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; log

_{2}

*p*&ceilr0; − 4 + 4p2

^{log 2p}, then

*f*(

*n*,

*p*,

*q*) ≤ e

^{Ologn }. This improves upon a result of Eichhorn and Mubayi. ^

#### Subject Area

Mathematics

#### Recommended Citation

Zachary A Kudlak,
"Problems in generalized graph colorings"
(2010).
*Dissertations and Master's Theses (Campus Access).*
Paper AAI3415496.

http://digitalcommons.uri.edu/dissertations/AAI3415496