#### Title

Labeling matched sums with a condition at distance two

#### Abstract

An *L(2,1)-labeling* of a graph *G *is a function *f*:* V*(*G*)* → *{0, 1,…, *k*}* *such that |*f*(*x*) – *f*(*y*)|* ≥ *2 if *x* and *y* are adjacent vertices, and |*f*(*x*) –* f*(*y*)|* *≥ 1 if *x* and *y* are at distance 2. The *lambda number* of *G* is the minimum *k* over all L(2,1)-labelings of *G*. Such labelings were introduced as a way of modeling the assignment of frequencies to transmitters operating in close proximity within a communications network. This paper considers the lambda number of the matched sum of two same-order disjoint graphs*, *wherein the graphs have been connected by a perfect matching between the two vertex sets. Matched sums have been studied in this context as a way of modeling possible connections between two different networks with the same number of transmitters. We completely determine the lambda number of matched sums where one of the graphs is a complete graph or a complete graph minus an edge. We conclude by discussing some difficulties that are encountered when trying to generalize this problem by removing more edges from a complete graph.

#### Academic Division

Math/Science

#### Disciplines

Applied Mathematics | Discrete Mathematics and Combinatorics

#### Recommended Citation

Troxell, Denise Sakai and Adams, Sarah Spence, "Labeling matched sums with a condition at distance two" (2010). *Babson Faculty Research Fund Working Papers*. 90.

https://digitalknowledge.babson.edu/bfrfwp/90

## Comments

Now published - for full text see:Troxell, D.S.; Adams, S.S. (2011) Labeling matched sums with a condition at distance two.

Applied Mathematics LettersVolume, 24(6), 950-957.http://dx.doi.org/doi:10.1016/j.aml.2011.01.004