[Paper Review] The Radio numbers of all graphs of order $n$ and diameter $n-2$

Katherine F. Benson, Matthew Porter+1 more|arXiv (Cornell University)|Jun 27, 2012

Graph Labeling and Dimension Problems3 citations

TL;DR

This paper determines the radio number for all graphs of order $n$ and diameter $n-2$, focusing on spire graphs $S_{n,s}$, which consist of a path of $n$ vertices with an additional edge from vertex $v_s$ to $v_n$. The authors establish exact formulas for the radio number based on parity of $n$ and the position $s$ of the spire, using novel labeling algorithms and lower-bound techniques. The key contribution is a complete classification of radio numbers for this family of graphs, resolving a gap left by prior work on paths ($n-1$ diameter).

ABSTRACT

A radio labeling of a connected graph $G$ is a function $c:V(G) o \mathbb Z_+$ such that for every two distinct vertices $u$ and $v$ of $G$ $$ ext{distance}(u,v)+|c(u)-c(v)|\geq 1+ ext{diameter}(G).$$ 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 $n-2$.

Motivation & Objective

• To determine the radio number for all connected graphs of order $n$ and diameter $n-2$, extending prior results on paths ($n-1$ diameter).
• To characterize the radio number of spire graphs $S_{n,s}$, defined as a path with an additional edge from $v_s$ to $v_n$, for all $2 \leq s \leq \lfloor n/2 \rfloor$.
• To establish tight upper and lower bounds for the radio number of spire graphs using constructive labeling and path-sum analysis.
• To generalize the results to all graphs of order $n$ and diameter $n-2$ by analyzing them as spire graphs with additional edges.
• To develop new techniques for deriving lower bounds on the radio number that are applicable to graphs with large diameter, including re-proving bounds for paths.

Proposed method

• Propose a labeling algorithm that assigns integer labels to vertices of $S_{n,s}$ to satisfy the radio condition: $d(u,v) + |c(u) - c(v)| \geq \text{diam}(G) + 1$.
• Use a vertex ordering strategy that groups vertices into three clusters based on distance from the spire and path symmetry to optimize label spacing.
• Define $c(x_i)$ as a piecewise function based on position in the label order, with step sizes derived from path distances and symmetry to minimize span.
• Establish upper bounds via explicit constructions for all cases: $n = 2k$ with $s \leq k-2$, $s = k-1$, $s = k$, and $n = 2k+1$.
• Derive lower bounds using path-sum maximization over all vertex orderings, applying Lemma 4 and edge contribution analysis via $N(e_i)$, the number of paths containing edge $e_i$.
• Use combinatorial bounds on $N(e_i)$ for edges $e_i$ in the path and the spire edge, with constraints from parity and path extremity conditions to tighten the lower bound.

Experimental results

Research questions

• RQ1 What is the radio number of all graphs of order $n$ and diameter $n-2$?
• RQ2 How does the position of the spire vertex $v_s$ affect the radio number of spire graphs $S_{n,s}$?
• RQ3 Can tight upper and lower bounds for the radio number be established for spire graphs using constructive labeling and path-sum analysis?
• RQ4 What new techniques can be developed to bound the radio number of graphs with large diameter, particularly when the diameter is $n-2$?
• RQ5 How do the results for spire graphs extend to all graphs of order $n$ and diameter $n-2$?

Key findings

• The radio number of $S_{n,s}$ is exactly $2k^2 - 4k + 2s + 3$ when $n = 2k$ and $2 \leq s \leq k-2$.
• For $n = 2k$ and $s = k-1$, the radio number is $2k^2 - 2k$.
• When $n = 2k$ and $s = k$, the radio number is $2k^2 - 2k + 1$.
• For odd $n = 2k+1$, the radio number of $S_{n,s}$ is $2k^2 - 2k + 2s$.
• The upper bounds from labeling constructions match the lower bounds derived from path-sum maximization, confirming the exact radio numbers for all cases.
• The results for spire graphs are used to determine the radio numbers of all other graphs of order $n$ and diameter $n-2$, which are either spire graphs with additional edges or variants thereof.