TY - JOUR

T1 - Independent sets in graphs with given minimum degree

AU - Alexander, James

AU - Cutler, Jonathan

AU - Mink, Tim

PY - 2012/9/27

Y1 - 2012/9/27

N2 - The enumeration of independent sets in graphs with various restrictions has been a topic of much interest of late. Let i(G) be the number of independent sets in a graph G and let it(G) be the number of independent sets in G of size t. Kahn used entropy to show that if G is an r-regular bipartite graph with n vertices, then i(G) ≤ i(Kr,r)n/2r. Zhao used bipartite double covers to extend this bound to general r-regular graphs. Galvin proved that if G is a graph with δ(G) ≥ δ and n large enough, then i(G) ≤ i(Kδ,n-δ). In this paper, we prove that if G is a bipartite graph on n vertices with δ(G) ≥ δ where n ≥ 2δ, then it(G) ≤ it(Kδ,n-δ) when t ≥ 3. We note that this result cannot be extended to t = 2 (and is trivial for t = 0, 1). Also, we use Kahn's entropy argument and Zhao's extension to prove that if G is a graph with n vertices, δ(G) ≥ δ, and Δ(G) ≤ Δ, then i(G) ≤ i(Kδ;≥)n/2δ.

AB - The enumeration of independent sets in graphs with various restrictions has been a topic of much interest of late. Let i(G) be the number of independent sets in a graph G and let it(G) be the number of independent sets in G of size t. Kahn used entropy to show that if G is an r-regular bipartite graph with n vertices, then i(G) ≤ i(Kr,r)n/2r. Zhao used bipartite double covers to extend this bound to general r-regular graphs. Galvin proved that if G is a graph with δ(G) ≥ δ and n large enough, then i(G) ≤ i(Kδ,n-δ). In this paper, we prove that if G is a bipartite graph on n vertices with δ(G) ≥ δ where n ≥ 2δ, then it(G) ≤ it(Kδ,n-δ) when t ≥ 3. We note that this result cannot be extended to t = 2 (and is trivial for t = 0, 1). Also, we use Kahn's entropy argument and Zhao's extension to prove that if G is a graph with n vertices, δ(G) ≥ δ, and Δ(G) ≤ Δ, then i(G) ≤ i(Kδ;≥)n/2δ.

UR - http://www.scopus.com/inward/record.url?scp=84867120323&partnerID=8YFLogxK

U2 - 10.37236/2722

DO - 10.37236/2722

M3 - Article

AN - SCOPUS:84867120323

SN - 1077-8926

VL - 19

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

IS - 3

ER -