Lower bounds on the independence number of a graph in terms of degrees
Jochen Harant, Ingo Schiermeyer
TL;DR
This paper addresses lower bounds on the independence number $\alpha(G)$ for graphs with bounded maximum degree $\Delta$, expressing bounds as linear combinations of the degree-class counts $|V_i(G)|$ with coefficients determined recursively. Building on the Kelly–Postle bound, it proves $\alpha(G) \ge \sum_{i=1}^{\Delta} c_i|V_i(G)|$ where $c_\Delta=1/\Delta$ and $i c_i + c_{i+1}=1$, and demonstrates that adding an additive term $\varepsilon|V_j(G)|$ is not universally valid. The authors introduce refined coefficients $d_i$ tied to the Euler number $e$, yielding sharper bounds in many cases, and provide concrete constructions (e.g., a graph $G^*$) to exhibit improvements over earlier bounds and to illustrate the tightness and noncomparability of different coefficient schemes. The work further establishes optimality aspects (Corollary C1, Theorem T3) and extends the framework with additional lower bounds, supported by detailed proofs in the accompanying section.
Abstract
Given an integer $Δ\ge 3$, let ${\cal G}_{Δ}$ be the set of connected graphs $G\neq K_{Δ+1}$ with maximum degree $Δ$ and, for $i=1,\cdots, Δ$, let $V_i(G)$ be the set of vertices of $G$ of degree $i$. Using a result of T. Kelly and L. Postle, we prove that $\sum\limits_{i=1}^Δc_i|V_i(G)|$ is a lower bound on the independence number $α(G)$ of $G\in {\cal G}_Δ$, where $c_Δ=\frac{1}Δ$ and $ic_{i}=1-c_{i+1}$ for $i=1,\cdots,Δ-1$. Moreover, if $\varepsilon >0$ and $j\in \{1,\cdots, Δ\}$, then the inequality $α(G)\ge \varepsilon|V_j(G)|+\sum\limits_{i=1}^Δc_i|V_i(G)|$ does not hold for infinitely many graphs $G\in {\cal G}_Δ$. Finally, further lower bounds on $α(G)$ in terms of degrees of $G$ are presented.