2 research outputs found

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

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    We study the problem of maximizing the number of independent sets in nn-vertex kk-chromatic β„“\ell-connected graphs. First we consider maximizing the total number of independent sets in such graphs with nn 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 nn-vertex kk-chromatic graphs with minimum degree at least β„“\ell, again for nn sufficiently large. We also maximize the number of independent sets of each fixed size in nn-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 nn-vertex kk-chromatic β„“\ell-connected graphs

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

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    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
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