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 .
"The $\Delta^2$ conjecture holds for graphs of small order." Involve 8 (4) 541 - 549, 2015. https://doi.org/10.2140/involve.2015.8.541