Lower bounds on the independence number of a graph in terms of degrees

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.

💡 Research Summary

The paper investigates lower bounds on the independence number α(G) of finite, simple, connected graphs G whose maximum degree is Δ≥3, excluding the complete graph K_{Δ+1}. For each i∈{1,…,Δ} let V_i(G) denote the set of vertices of degree i. The authors build on a recent result by Kelly and Postle, which states that if a weight function g:V(G)→ℝ satisfies g(v)≤2/(2d(v)+1) for every vertex v and ∑{v∈K}g(v)≤1 for every clique K, then α(G)≥∑{v∈V(G)}g(v).

Choosing g(v)=c_i whenever v∈V_i(G) leads to a system of coefficients {c_i} defined by
c_Δ = 1/Δ,
i·c_i + c_{i+1} = 1 for i=1,…,Δ−1.
These equations uniquely determine a decreasing sequence 0<c_i<1. Because any clique in G has size at most Δ, the clique condition reduces to i·c_i + c_{i+1}=1, which holds by construction. Consequently the Kelly‑Postle theorem yields the fundamental bound

 α(G) ≥ Σ_{i=1}^{Δ} c_i·|V_i(G)| .  (2)

For Δ=3 this reproduces the known inequality α(G) ≥ (2/3)|V_1| + (1/3)|V_2| + (1/3)|V_3|. The authors prove (2) for all Δ≥4 by induction on |V(G)|, using two lemmas. Lemma 2 shows that removing a minimum‑degree vertex u together with its neighbours reduces the problem to the isolated vertices of the remaining graph and to each connected component H of the remainder; the contribution of u and its neighbours is exactly c_{d(u)} + Σ_{w∈N(u)}c_{d(w)} = 1. Lemma 3 establishes that any proper subgraph H with maximum degree Δ′<Δ satisfies the same bound with the corresponding subset of coefficients. Combining these lemmas yields the inductive step.

The paper then addresses optimality of the coefficients. Theorem 3 states that for any ε>0 and any index j, the stronger inequality

 α(G) ≥ ε·|V_j(G)| + Σ_{i=1}^{Δ} c_i·|V_i(G)|

fails for infinitely many graphs in the family. The authors construct infinite families of Δ‑regular graphs in which the independence number equals exactly Σ c_i·|V_i|, thereby showing that no positive ε can be added for any degree class. The construction uses copies of K_Δ attached via a perfect matching to a Δ‑regular base graph, preserving regularity while keeping α(G) minimal.

Further refinements are provided. Theorem 4 considers a graph G whose maximum degree is Δ′<Δ (i.e., a proper subgraph of a Δ‑regular graph). By embedding G into a Δ‑regular graph with a few extra pendant edges, the same coefficient sequence yields the bound

 α(G) ≥ Σ_{i=1}^{Δ′} c_i·|V_i(G)| .

Because the coefficients depend on Δ, the bound for Δ′ can be either stronger or weaker than the bound obtained directly with Δ′’s own coefficient sequence; the two bounds are not comparable in general.

Theorem 5 introduces an alternative set of coefficients {d_i} that are easier to compute for large Δ. They are defined via a closed form involving the Euler number e and factorials:

 d_1 = 1 – 1/e, d_{i+1} = 1 – i·d_i (i≥1).

Explicitly,

 d_i = 1/(i+1) + (i−1)!·