Extremal Colorings and Independent Sets

Abstract

This paper is NOT THE PUBLISHED VERSION; but the author's final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation below.

Independent Sets in n-vertex k-chromatic, \ell-connected graphs

“…This focus of this note is on maximizing i(G) and i t (G) within the family of n-vertex k-chromatic ℓ-connected graphs. When ℓ = 0 and ℓ = 1, the maximum number of independent sets, and independent sets of each fixed size t, in these families was determined in [5]. Our first result generalizes this to maximizing i(G) when ℓ > 1 for n large.…”

“…These last two conjectures appeared as questions in [5]. We note that the corresponding results for k = 3 and ℓ = 2 are shown in Theorem 1.5, and that the cases when ℓ = 0 and ℓ = 1 appear in [5].…”

“…So, for example, when k = 2 ≤ ℓ we have G * = K ℓ,n−ℓ , and the fact that i(G) ≤ i(G * ) for all n-vertex ℓ-connected bipartite graphs G, with equality if and only if G = G * , appears as Corollary 2.2 in [1] (recall that an ℓ-connected graph has minimum degree at least ℓ). When ℓ = 1 and k > 1 the graph G * is formed from a k-clique with n − k pendants attached to one vertex in the clique, and the fact that i(G) ≤ i(G * ) for all connected k-chromatic graphs G, with equality if and only if G = G * , appears as Corollary 3 in [5]. (Viewing G * naturally as K k ∪ E n−k in the case where ℓ = 0, the analogous result appears as Corollary 2 in [5].…”

“…When ℓ = 1 and k > 1 the graph G * is formed from a k-clique with n − k pendants attached to one vertex in the clique, and the fact that i(G) ≤ i(G * ) for all connected k-chromatic graphs G, with equality if and only if G = G * , appears as Corollary 3 in [5]. (Viewing G * naturally as K k ∪ E n−k in the case where ℓ = 0, the analogous result appears as Corollary 2 in [5]. )…”

“…In [5], the authors ask what can be said in the family of n-vertex k-chromatic ℓ-connected graphs for independent sets of size t when ℓ > 1. We also provide some results here for specific values of k, ℓ, and t. The first new case is that of 2-connected 3-chromatic graphs; we remark that results on maximizing i(G) over all n-vertex 2-connected graphs appear in [12].…”