On the k-residue of disjoint unions of graphs with applications to k-independence

The k-residue of a graph, introduced by Jelen in a 1999 paper, is a lower bound on the k-independence number for every positive integer k. This generalized earlier work by Favaron, Mahéo, and Saclé, by Griggs and Kleitman, and also by Triesch, who all showed that the independence number of a graph is at least as large as its Havel-Hakimi residue, defined by Fajtlowicz. We show here that, for every positive integer k, the k-residue of disjoint unions is at most the sum of the k-residues of the connected components considered separately, and give applications of this lemma. Our main application is an improvement on Jelen’s bound for connected graphs which have a maximum degree cut-vertex. We demonstrate constructively that, in some cases, our extensions give better approximations to the k-independence number than all known lower bounds – including bounds of Hopkins and Staton, Caro and Tuza, Favaron, Caro and Hansberg, as well as Jelen’s k-residue bound itself. Additionally, we apply this disjoint union lemma to prove a theorem for function graphs (those graphs formed by connecting vertices from a graph and its copy according to a given function) while simultaneously giving, in this context, different classes of non-trivial examples for which our new results improve on the k-residue, further motivating our first application of the lemma.