# The Maximum Number of Complete Subgraphs in a Graph with Given Maximum Degree

## Document Type

Article

## Publication Date

1-1-2014

## Abstract

Extremal problems involving the enumeration of graph substructures have a long history in graph theory. For example, the number of independent sets in a d-regular graph on n vertices is at most (2d+1-1)n/2d by the Kahn-Zhao theorem [7,13]. Relaxing the regularity constraint to a minimum degree condition, Galvin [5] conjectured that, for n≥2d, the number of independent sets in a graph with δ(G)≥d is at most that in Kd,n-d.In this paper, we give a lower bound on the number of independent sets in a d-regular graph mirroring the upper bound in the Kahn-Zhao theorem. The main result of this paper is a proof of a strengthened form of Galvin's conjecture, covering the case n≤2d as well. We find it convenient to address this problem from the perspective of G-. From this perspective, we show that the number of complete subgraphs of a graph G on n vertices with δ(G)≤r, where n=a(r+1)+b with 0≤b≤r, is bounded above by the number of complete subgraphs in aKr+1∪Kb.

## DOI

10.1016/j.jctb.2013.10.003

## MSU Digital Commons Citation

Cutler, Jonathan and Radcliffe, A. J., "The Maximum Number of Complete Subgraphs in a Graph with Given Maximum Degree" (2014). *Department of Mathematics Facuty Scholarship and Creative Works*. 173.

https://digitalcommons.montclair.edu/mathsci-facpubs/173