[Paper Review] On cycles in graphs with specified radius and diameter

P. HRNČIAR|arXiv (Cornell University)|Jul 2, 2012

Graph theory and applications2 references3 citations

TL;DR

This paper establishes a tight lower bound on the circumference of a graph in terms of its radius $ r $ and diameter $ d $, proving that any graph with $ d \leq 2r - 2 $ must contain a cycle of length at least $ 4r - 2d $. The result is sharp, as shown by constructing explicit families of graphs—such as sun-graphs $ S_{4r-2d, d-r} $—that achieve this bound for all valid $ r $ and $ d $, resolving a long-standing open question about cycle length in graphs with specified radius and diameter.

ABSTRACT

Let $G$ be a graph of radius $r$ and diameter $d$ with $d\leq 2r-2$. We show that $G$ contains a cycle of length at least $4r-2d$, i.e. for its circumference it holds $c(G)\geq 4r-2d$. Moreover, for all positive integers $r$ and $d$ with $r\leq d\leq 2r-2$ there exists a graph of radius $r$ and diameter $d$ with circumference $4r-2d$.

Motivation & Objective

• To resolve an open problem regarding the minimum cycle length in graphs with specified radius and diameter, particularly for $ d \leq 2r - 2 $.
• To establish a tight lower bound on the circumference $ c(G) $ in terms of radius $ r $ and diameter $ d $.
• To construct explicit families of graphs achieving the bound $ c(G) = 4r - 2d $, proving its sharpness.
• To generalize prior results on geodesic cycles and minimal-order graphs with given radius and diameter.

Proposed method

• Using proof by contradiction, the paper assumes $ c(G) < 4r - 2d $ and analyzes the structure of blocks and cut-vertices in the graph.
• It distinguishes cases based on the distance from a vertex of maximum eccentricity to a central block $ B $, considering distances $ a \leq r-1 $ and $ a \geq r $.
• The analysis leverages geodesic paths and cycles, particularly focusing on vertices and paths that would violate the radius constraint if the circumference were too small.
• It employs the concept of sun-graphs $ S_{m,k} $, which are unicyclic graphs with degree constraints, to construct extremal examples achieving the bound.
• The proof iteratively reduces the graph by replacing subgraphs to derive a contradiction, showing that the initial assumption $ c(G) < 4r - 2d $ cannot hold.
• The construction of extremal graphs uses $ S_{4r-2d, d-r} $, a sun-graph with cycle length $ 4r - 2d $ and $ 4r - 2d $ rays of length $ d - r $, to demonstrate tightness of the bound.

Experimental results

Research questions

• RQ1 What is the minimum possible circumference of a graph with radius $ r $ and diameter $ d $, where $ d \leq 2r - 2 $?
• RQ2 Can a tight lower bound on the circumference be established in terms of $ r $ and $ d $?
• RQ3 Are there extremal graphs that achieve this bound, and if so, what are their structural properties?
• RQ4 How does the bound relate to known results on geodesic cycles and minimal-order graphs?

Key findings

• The paper proves that any graph $ G $ with radius $ r $ and diameter $ d \leq 2r - 2 $ must have a circumference $ c(G) \geq 4r - 2d $.
• The bound $ c(G) \geq 4r - 2d $ is tight, as demonstrated by the existence of graphs achieving equality.
• For all integers $ r $ and $ d $ with $ r \leq d \leq 2r - 2 $, there exist infinitely many graphs with radius $ r $, diameter $ d $, and circumference $ 4r - 2d $.
• One such extremal graph is the cycle $ C_{2r} $ when $ d = r $, and for $ d > r $, the sun-graph $ S_{4r-2d, d-r} $ achieves the bound.
• The result implies that graphs with radius $ r $ and diameter $ d \leq 2r - 2 $ must contain cycles of length at least 4, as $ 4r - 2d \geq 4 $ when $ d \leq 2r - 2 $.
• The bound $ 3r - 2 $ on the number of vertices for which $ c(G) \geq 2r $ is shown to be best possible, as counterexamples exist with $ 3r - 1 $ vertices and $ c(G) = 2r - 1 $.