TL;DR
This paper explores the application of the Kruskal-Katona theorem to graphs, identifying cases where the bound is tight or not, and discusses techniques for counting complete subgraphs.
Contribution
It extends the known cases where the Kruskal-Katona bound is tight and investigates methods for accurately counting complete subgraphs in graphs.
Findings
Identified new cases where the Kruskal-Katona bound is tight
Analyzed instances where the bound is not tight
Presented techniques for counting complete subgraphs
Abstract
In graph theory, knowing the number of complete subgraphs with r vertices that a graph g has, limits the number of its complete subgraphs with s vertices, for s > r. A useful upper bound is provided by the Kruskal-Katona theorem, but this bound is often not tight. In this note, we add to the known cases where this bound is tight and also investigate cases where it is not. Finally we look at a useful technique for actually finding the numbers of complete subgraphs of a graph.
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