## Internet Mathematics

### Real Number Labelings for Paths and Cycles

#### Abstract

The problem of radio channel assignments with multiple levels of interference depending on distance can be modelled using graph theory. The authors previously introduced a model of labeling by real numbers. Given a graph $G$, possibly infinite, and real numbers $k_1,k_2\ge0$, an $L(k_1,k_2)$-labeling of $G$ assigns real numbers $f(x)\ge0$ to the vertices $x$, such that the labels of vertices $u$ and $v$ differ by at least $k_i$ if $u$ and $v$ are at distance $i$ apart. We denote by $\lambda(G;k_1,k_2)$ the infimum span over such labelings~$f$. It is enough to determine $\lambda(G;k,1)$ for reals $k\ge0$, which will be a continuous nondecreasing piecewise linear function. Here we present these functions for paths, cycles, and wheels.

#### Article information

Source
Internet Math., Volume 4, Number 1 (2007), 65-86.

Dates
First available in Project Euclid: 27 May 2009