TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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An $n$ -vertex graph of minimum degree $δ > 1$ has a dominating set of size at most $n (\log (δ + 1) + 1) / (δ + 1)$

Statement

Let $G$ be an $n$ -vertex graph with minimum degree $δ > 1$ . Then $γ (G) \leq \frac{n (\log (δ + 1) + 1)}{δ + 1} .$

Facts & Assumptions

Given: An $n$ -vertex graph $G$ of minimum degree $δ > 1$ .

[L1]

A dominating set contains or neighbours every vertex (Dominating sets in a finite graph).

[L2]

Independent Bernoulli coordinate choices form a finite product space (Product weights normalize, and coordinate events are mutually independent).

[L3]

Expectation is linear and some outcome is at most its expectation (Expectation is linear for every finite family of random variables, without any independence hypothesis, Expectation preserves pointwise order and lies between the minimum and maximum attained values).

[L4]

$(1 - p)^{m} \leq \exp (- m p)$ for $0 \leq p \leq 1$ ( $1 + x \leq \exp (x)$ for every real $x$ , hence $(1 - p)^{m} \leq \exp (- m p)$ ).

[L5]

The natural logarithm is increasing and satisfies its inverse and product laws (The natural logarithm as the inverse of the exponential function, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm).

Proof

technique · constructive

1.1

Put $p = \log (δ + 1) / (δ + 1)$ . Since $δ + 1 > 1$ and the logarithm is increasing with $\log 1 = 0$ , one has $p > 0$ . Applying [L4] at $- p$ is not needed here: applying its first inequality at $\log (δ + 1)$ gives $1 + \log (δ + 1) \leq δ + 1$ , hence $p \leq δ / (δ + 1) < 1$ .

L4L5algebra

1.2

Select every vertex independently with probability $p$ , obtaining $S$ , and let $U$ be the vertices neither in $S$ nor adjacent to a member of $S$ . Then $D = S \cup U$ is dominating.

L1L2construct

2.1

A fixed vertex belongs to $U$ only if none of at least $δ + 1$ vertices in its closed neighbourhood is selected, so $P (v \in U) \leq (1 - p)^{δ + 1} \leq \exp (- p (δ + 1)) = 1 / (δ + 1)$ .

step 1.1step 1.2L2L4L5

3.1

By linearity, $E [∣ D ∣] \leq n p + n / (δ + 1) = n (\log (δ + 1) + 1) / (δ + 1)$ .

step 2.1L3algebra

4.1

Some outcome has $∣ D ∣$ at most this expectation, and its $D$ is a dominating set by step 1.2.

step 1.2step 3.1L3discharge-construct∎

Depends on

Used by

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Sources

Y. Zhao, MIT 18.218 Probabilistic Method in Combinatorics, Theorem 6.1 (standard reference, not scraped)

An nn-vertex graph of minimum degree δ>1δ>1 has a dominating set of size at most n(log⁡(δ+1)+1)/(δ+1)n(log(δ+1)+1)/(δ+1)