Vol. 8, No. 4, 2015

Download this article
Download this article For screen
For printing
Recent Issues

Volume 12
Issue 5, 721–899
Issue 4, 541–720
Issue 3, 361–539
Issue 2, 181–360
Issue 1, 1–180

Volume 11, 5 issues

Volume 10, 5 issues

Volume 9, 5 issues

Volume 8, 5 issues

Volume 7, 6 issues

Volume 6, 4 issues

Volume 5, 4 issues

Volume 4, 4 issues

Volume 3, 4 issues

Volume 2, 5 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Scientific Advantages
Submission Guidelines
Submission Form
Ethics Statement
Editorial Login
Author Index
Coming Soon
ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
Other MSP Journals
This article is available for purchase or by subscription. See below.
The $\Delta^2$ conjecture holds for graphs of small order

Cole Franks

Vol. 8 (2015), No. 4, 541–549

An L(2,1)-labeling of a simple graph G is a function f : V (G) 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 (Δ2 + 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.

PDF Access Denied

However, your active subscription may be available on Project Euclid at

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 30.00:

L(2,1)-labeling, graph labeling, channel assignment
Mathematical Subject Classification 2010
Primary: 97K30
Received: 15 February 2013
Revised: 15 April 2013
Accepted: 2 October 2013
Published: 23 June 2015

Communicated by Ronald Gould
Cole Franks
Department of Mathematics
Rutgers University
Hill Center - Busch Campus
110 Frelinghuysen Road
Piscataway, NJ 08854
United States