Involve: A Journal of Mathematics
- Volume 8, Number 4 (2015), 541-549.
The $\Delta^2$ conjecture holds for graphs of small order
An -labeling of a simple graph is a function such that if , then , and if the distance between and is two, then . -labelings are motivated by radio channel assignment problems. Denote by the smallest integer such that there exists an -labeling of using the integers . We prove that , where , if the order of is no greater than . This shows that for graphs no larger than the given order, the 1992 “ conjecture” of Griggs and Yeh holds. In fact, we prove more generally that if , , and
then . In addition, we exhibit an infinite family of graphs with .
Involve, Volume 8, Number 4 (2015), 541-549.
Received: 15 February 2013
Revised: 15 April 2013
Accepted: 2 October 2013
First available in Project Euclid: 22 November 2017
Permanent link to this document
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Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 97K30: Graph theory
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