More Cases Where the Kruskal-Katona Bound is Tight

TL;DR

This paper explores specific cases where the Kruskal-Katona bound on the number of complete subgraphs in a graph is tight, and initiates the study of sequences of graphs with tight bounds.

Contribution

It identifies new instances where the Kruskal-Katona bound is tight and starts analyzing sequences of graphs that achieve these bounds.

Findings

New cases where the Kruskal-Katona bound is tight

Initial results on tight bounded graph sequences

Enhanced understanding of subgraph count bounds

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 begin an investigation of tight bounded graph sequences.