[Paper Review] On wheel-free graphs

Pierre Aboulker, Frédéric Havet+1 more|arXiv (Cornell University)|Sep 9, 2013

Advanced Graph Theory Research14 references3 citations

TL;DR

This paper investigates the structural properties of wheel-free graphs—graphs that do not contain a wheel (a chordless cycle with a universal vertex) as a subgraph. It proves that every 3-connected wheel-free graph is minimally 3-connected and uses this to show that such graphs have a vertex of degree at most 3. The key contribution is a new proof of the 3-colorability of wheel-free graphs, extending results on Berge graphs and structural graph theory.

ABSTRACT

A wheel is a graph formed by a chordless cycle and a vertex that has at least three neighbors in the cycle. We prove that every 3-connected graph that does not contain a wheel as a subgraph is in fact minimally 3-connected. We give a new proof of a theorem of Thomassen and Toft: every graph that does not contain a wheel as a subgraph is 3-colorable.

Motivation & Objective

• To understand the structural properties of graphs that do not contain a wheel as a subgraph.
• To investigate whether wheel-free graphs have bounded chromatic number, addressing an open question on c-colorability.
• To establish that 3-connected wheel-free graphs are minimally 3-connected, implying a vertex of degree at most 3.
• To provide a new proof of the 3-colorability of wheel-free graphs, building on Thomassen and Toft's result.
• To explore the implications of excluding wheels for graph coloring and connectivity, particularly in relation to Truemper configurations.

Proposed method

• Uses Menger’s Theorem and variants to analyze connectivity and paths in wheel-free graphs.
• Applies induction and structural decomposition to analyze blocks and pairs of twins in 3-connected graphs.
• Employs a new proof technique for the theorem of Watkins and Mesner on cycles through three vertices.
• Leverages the minimality of 3-connected wheel-free graphs to derive degree bounds and colorability results.
• Analyzes subgraphs isomorphic to $K_{3,3} \setminus e$ and $K_{3,3}$ to deduce structural constraints.
• Uses case analysis on edge presence between vertex pairs in 3-connected components to derive twin pair existence.

Experimental results

Research questions

• RQ1 Is every 3-connected wheel-free graph minimally 3-connected?
• RQ2 Do all wheel-free graphs have a vertex of degree at most 3?
• RQ3 Can the 3-colorability of wheel-free graphs be reproven using structural arguments?
• RQ4 What is the chromatic number of graphs that exclude long wheels (rim length ≥4) as subgraphs?
• RQ5 How do the exclusion of wheels and subdivisions of $K_4$ affect the structural and coloring properties of graphs?

Key findings

• Every 3-connected wheel-free graph is minimally 3-connected, a structural result that implies the existence of a vertex of degree at most 3.
• Every wheel-free graph has a vertex of degree at most 3, confirming a special case of Turner’s theorem.
• Every planar wheel-free graph has a vertex of degree at most 2, extending the degree bound to planar graphs.
• A new, shorter proof is given for the theorem of Watkins and Mesner on the non-existence of cycles through three vertices under certain connectivity conditions.
• Any 3-connected wheel-free graph contains two disjoint pairs of twins, a structural feature used in the proof of minimality and colorability.
• Graphs that exclude long wheels (rim length ≥4) as subgraphs are 4-colorable, and if they are also wheel-free, they are 3-colorable.