Vol. 8, No. 4, 2015

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

Volume 11, 1 issue

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
Cover Page
Editorial Board
Editors’ Addresses
Editors’ Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Ethics Statement
Subscriptions
Editorial Login
Author Index
Coming Soon
Contacts
 
ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
The $\Delta^2$ conjecture holds for graphs of small order

Cole Franks

Vol. 8 (2015), No. 4, 541–549
Abstract

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.

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

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