## Involve: A Journal of Mathematics

• Involve
• Volume 8, Number 4 (2015), 541-549.

### The $\Delta^2$ conjecture holds for graphs of small order

Cole Franks

#### Abstract

An $L(2,1)$-labeling of a simple graph $G$ is a function $f : V (G) → ℤ$ such that if $xy ∈ E(G)$, then $|f(x) − f(y)|≥ 2$, and if the distance between $x$ and $y$ is two, then $|f(x) − f(y)|≥ 1$. $L(2,1)$-labelings are motivated by radio channel assignment problems. Denote by $λ2,1(G)$ the smallest integer such that there exists an $L(2,1)$-labeling of $G$ using the integers ${0,…,λ2,1(G)}$. We prove that $λ2,1(G) ≤ Δ2$, where $Δ = Δ(G)$, if the order of $G$ is no greater than $(⌊Δ∕2⌋ + 1)(Δ2 − Δ + 1) − 1$. This shows that for graphs no larger than the given order, the 1992 “$Δ2$ conjecture” of Griggs and Yeh holds. In fact, we prove more generally that if $L ≥ Δ2 + 1$, $Δ ≥ 1$, and

$|V (G)|≤ (L − Δ)(⌊L − 1 2Δ ⌋ + 1) − 1,$

then $λ2,1(G) ≤ L − 1$. In addition, we exhibit an infinite family of graphs with $λ2,1(G) = Δ2 − Δ + 1$.

#### Article information

Source
Involve, Volume 8, Number 4 (2015), 541-549.

Dates
Revised: 15 April 2013
Accepted: 2 October 2013
First available in Project Euclid: 22 November 2017

https://projecteuclid.org/euclid.involve/1511370909

Digital Object Identifier
doi:10.2140/involve.2015.8.541

Mathematical Reviews number (MathSciNet)
MR3366009

Zentralblatt MATH identifier
1316.05106

Subjects
Primary: 97K30: Graph theory

#### Citation

Franks, Cole. The $\Delta^2$ conjecture holds for graphs of small order. Involve 8 (2015), no. 4, 541--549. doi:10.2140/involve.2015.8.541. https://projecteuclid.org/euclid.involve/1511370909

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