Rainbow Spanning Subgraphs of Small Diameter in Edge-Colored Complete Graphs

Abstract

Let s(n, t) be the maximum number of colors in an edge-coloring of the complete graph $K_n$ that has no rainbow spanning subgraph with diameter at most t. We prove $s(n,t)={\left( {\begin{array}{c}n-2\\ 2\end{array}}\right) }+1$ for $n,t\ge 3$, while $s(n,2)={\left( {\begin{array}{c}n-2\\ 2\end{array}}\right) }+\left\lfloor {\frac{n-1}{2}}\right\rfloor$ for $n\ne 4$ (and $s(4,2)=2$).