Counting Independent Sets of a Fixed Size in Graphs with a Given Minimum Degree

Abstract

Galvin showed that for all fixed δ and sufficiently large n, the n-vertex graph with minimum degree δ that admits the most independent sets is the complete bipartite graph K δ,n−δ . He conjectured that except perhaps for some small values of t, the same graph yields the maximum count of independent sets of size t for each possible t. Evidence for this conjecture was recently provided by Alexander, Cutler, and Mink, who showed that for all triples (n, δ, t) with t ≥ 3, no n-vertex bipartite graph with minimum degree δ admits more independent sets of size t than K δ,n−δ .Here we make further progress. We show that for all triples (n, δ, t) with δ ≤ 3 and t ≥ 3, no n-vertex graph with minimum degree δ admits more independent sets of size t than K δ,n−δ , and we obtain the same conclusion for δ > 3 and t ≥ 2δ + 1. Our proofs lead us naturally to the study of an interesting family of critical graphs, namely those of minimum degree δ whose minimum degree drops on deletion of an edge or a vertex.

Extremal H‐Colorings of Graphs with Fixed Minimum Degree

“…Recently, Cutler and Radcliffe [5] have extended Theorem 1.2 to the range n ≥ 2δ. Further results related to maximizing the number of independent sets of a fixed size for G ∈ G(n, δ) can be found in, for example, [1,2,7,18], with a complete answer to this question for n ≥ 2δ given by Gan et al [15].…”

Extremal Regular Graphs: Independent Sets and Graph Homomorphisms

“…The following strengthening was conjectured by Engbers and Galvin [26]. It was proved by Alexander, Cutler, and Mink [1] for bipartite graphs, and proved by Gan, Loh, and Sudakov [36] in general.…”

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

“…Further, Galvin conjectured that the same graph has the largest number of independent sets of each size , even though it has fewer independent sets of size two than a d ‐regular graph. Engbers and Galvin proved this stronger conjecture for . Other progress on the stronger conjecture was made by Law and McDiarmid and Alexander and Mink .…”

“…The case of this theorem corresponds (in the complement) to Galvin's original conjecture (i.e., ). The case was proved independently by Engbers and Galvin and Wood . What now remains is the level set version to Theorem , conjectured by Gan et al.…”

“…For all n, r ∈ N, write n = a(r + 1) + b with 0 ≤ b ≤ r. If G is a graph on n vertices with (G) ≤ r, thenwith equality if and only if G = aK r+1 ∪ K b , or r = 2 and G = (a − 1)K 3 ∪ C 4 or (a − 1)K 3 ∪ C 5 .The case a = 1 of this theorem corresponds (in the complement) to Galvin's original conjecture (i.e., n ≥ 2d). The case b = 0 was proved independently by Engbers and Galvin [5] and Wood [10]. What now remains is the level set version to Theorem 1.2, conjectured by Gan et alMain Conjecture.…”