Cardinalities of the total number of independent sets

Abstract

We study the set of numbers the total number of independent sets can admit in $n$-vertex graphs. In this paper, we prove that the cardinality $\mathcal{N}i(n)$ of this set is very close to $2^n$ in the following sense: $\mathcal{N}i(n)/2^n = O(n^{-1/5})$ while for infinitely many $n$, we have $\log_2(\mathcal{N}i(n)/2^n)\ge -2^{(1+o(1)\sqrt{\log_2 n}}$. This set is also precisely the set of possible values of the independence polynomial $I_G(x)$ at $x=1$ for $n$-vertex graphs $G$. As an application, we address an additive combinatorial problem on subsets of a given vector space that avoid certain intersection patterns with respect to subspaces.

Figures (2)

• Figure 1: An example for a partial join of $G_L=\overline{K_{15}}$ and $G_R=4K_3$. For the red, blue and green vertices of $G_R$, their neighbour sets in $G_L$ are denoted with identically coloured boxes, whereas the rest of the vertices of $G_R$ have no neighbours in $G_L$. The partial transversal $T_R=\{w_{1,1}, w_{3,2}, w_{4,1}\}$ consisting of the three coloured vertices has $|V_L\setminus \cup_{v_R\in T_R} N_L(v_R)|=3$, therefore it contributes $i(\overline{K_3})=8$ to the total sum $i(G)$ shown in \ref{['eq:disjointtrick']}. In contrast, the full transversal $T'_R=\{w_{1,1}, w_{2,1}, w_{3,1}, w_{4,1}\}$ would contribute $2^4=16$.
• Figure 2: An example for $d=2$ and $m=4$, with the subset $S\subseteq Q$ shown in dark blue, and the additional points extending $S$ to $S_c$ shown in light blue. The parts of $\widehat{Q}$ belonging to each vector $\mathbf{v}$ are coloured according to the parity of $\mathbf{v}$.

Theorems & Definitions (33)

• Definition 1.1
• Theorem 1.3
• Lemma 2.1
• Proposition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 3.2: Oruc
• Corollary 3.3
• Corollary 3.4
• proof