The Radio numbers of all graphs of order $n$ and diameter $n-2$
Abstract
A radio labeling of a connected graph $G$ is a function $c:V(G) \to \mathbb Z_+$ such that for every two distinct vertices $u$ and $v$ of $G$ $$\text{distance}(u,v)+|c(u)-c(v)|\geq 1+ \text{diameter}(G).$$ The radio number of a graph $G$ is the smallest integer $M$ for which there exists a labeling $c$ with $c(v)\leq M$ for all $v\in V(G)$. The radio number of graphs of order $n$ and diameter $n-1$, i.e., paths, was determined in \cite{paths}. Here we determine the radio numbers of all graphs of order $n$ and diameter $n-2$.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2012
- DOI:
- 10.48550/arXiv.1206.6327
- arXiv:
- arXiv:1206.6327
- Bibcode:
- 2012arXiv1206.6327B
- Keywords:
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- Mathematics - Combinatorics;
- 05C78 (05C15;
- 05C38)
- E-Print:
- 21 pages, 10 figures