On edge colorings with at least \(q\) colors in every subset of \(p\) vertices

Gábor N. Sárközy ; Stanley Selkow

The electronic journal of combinatorics, Tome 8 (2001) no. 1

Résumé

For fixed integers $p$ and $q$, an edge coloring of $K_n$ is called a $(p, q)$-coloring if the edges of $K_n$ in every subset of $p$ vertices are colored with at least $q$ distinct colors. Let $f(n, p, q)$ be the smallest number of colors needed for a $(p, q)$-coloring of $K_n$. In [3] Erdős and Gyárfás studied this function if $p$ and $q$ are fixed and $n$ tends to infinity. They determined for every $p$ the smallest $q$ ($= {p \choose 2} - p + 3$) for which $f(n,p,q)$ is linear in $n$ and the smallest $q$ for which $f(n,p,q)$ is quadratic in $n$. They raised the question whether perhaps this is the only $q$ value which results in a linear $f(n,p,q)$. In this paper we study the behavior of $f(n,p,q)$ between the linear and quadratic orders of magnitude. In particular we show that that we can have at most $\log p$ values of $q$ which give a linear $f(n,p,q)$.

Détail

DOI : 10.37236/1553

Classification : 05C15, 05C55
Mots-clés : edge coloring, monochromatic matching, probabilistic upper bound