TY - JOUR

T1 - The maximum number of complete subgraphs in a graph with given maximum degree

AU - Cutler, Jonathan

AU - Radcliffe, A. J.

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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.

AB - 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.

KW - Complete subgraphs

KW - Extremal enumeration

KW - Independent sets

UR - http://www.scopus.com/inward/record.url?scp=84888380405&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2013.10.003

DO - 10.1016/j.jctb.2013.10.003

M3 - Article

AN - SCOPUS:84888380405

VL - 104

SP - 60

EP - 71

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 1

ER -