Independent Sets in n-vertex k-chromatic, \ell-connected graphs

We study the problem of maximizing the number of independent sets in $n$-vertex $k$-chromatic $\ell$-connected graphs. First we consider maximizing the total number of independent sets in such graphs with $n$ sufficiently large, and for this problem we use a stability argument to find the unique extremal graph. We show that our result holds within the larger family of $n$-vertex $k$-chromatic graphs with minimum degree at least $\ell$, again for $n$ sufficiently large. We also maximize the number of independent sets of each fixed size in $n$-vertex 3-chromatic 2-connected graphs. We finally address maximizing the number of independent sets of size 2 (equivalently, minimizing the number of edges) over all $n$-vertex $k$-chromatic $\ell$-connected graphs

An Improved Bound on the Minimal Number of Edges in Color-Critical Graphs

It is proven that for k # 4 and n > k every k-color-critical graph on n vertices has at least # k-1 2 + k-3 2(k 2 -2k-1) # n edges, thus improving a result of Gallai from 1963