[Paper Review] Approximating Vizing's independence number conjecture

Eckhard Steffen|arXiv (Cornell University)|Sep 3, 2016

Advanced Graph Theory Research3 citations

TL;DR

This paper proves that Vizing's 1965 conjecture — that the independence ratio of edge-chromatic critical graphs is at most 1/2 — is equivalent to its restriction on a specific family of critical graphs with independence ratio strictly less than 1/2 + ε for any ε > 0. Using Meredith extensions and structural analysis of k-critical graphs with distinguished paths, the authors show that the independence ratio of these graphs approaches 1/2 as path length increases, thereby reducing the full conjecture to a finite, ε-approximate verification problem.

ABSTRACT

In 1965, Vizing conjectured that the independence ratio of edge-chromatic critical graphs is at most $\frac{1}{2}$. We prove that for every $ε> 0$ this conjecture is equivalent to its restriction on a specific set of edge-chromatic critical graphs with independence ratio smaller than $\frac{1}{2} + ε$.

Motivation & Objective

• To resolve Vizing's long-standing conjecture that the independence ratio of edge-chromatic critical graphs is at most 1/2.
• To show that verifying the conjecture on a specific, structured subclass of critical graphs is sufficient to prove it in general.
• To establish that the independence ratio of k-critical graphs in a defined subclass approaches 1/2 as structural parameters grow.
• To provide a framework for ε-approximate verification of the conjecture by restricting attention to graphs with independence ratio bounded above by 1/2 + ε.

Proposed method

• Introduces the class 𝒞(k,t) of k-critical graphs with vertices of degree k−1 that each initiate k−1 vertex-disjoint, long paths of degree k−1 vertices.
• Uses Meredith extensions to construct new k-critical graphs from existing ones, preserving criticality and enabling recursive construction.
• Applies Vizing’s Adjacency Lemma to ensure structural constraints on neighbors of low-degree vertices in k-critical graphs.
• Derives an upper bound on the independence ratio of graphs in 𝒞(k,t) using path length and vertex-disjointness constraints: ι(G) < 1/2 + 1/(4kφ(k,t) + 2), where φ(k,t) = t(k−1)² + k−1.
• Proves that limₜ→∞ ι(k,t) = 1/2, showing the bound becomes arbitrarily tight as path length increases.
• Constructs a universal family 𝒞_ε = ⋃ₖ≥₂ 𝒞(k,t₃) for a chosen t₃ such that all graphs in 𝒞_ε have independence ratio < 1/2 + ε.

Experimental results

Research questions

• RQ1 Is Vizing’s conjecture on the independence ratio of edge-chromatic critical graphs equivalent to its restriction on a specific subclass of graphs with independence ratio bounded by 1/2 + ε?
• RQ2 Can the independence ratio of k-critical graphs be uniformly bounded away from 1/2 in a structured subclass, and does it approach 1/2 as structural parameters grow?
• RQ3 Does the use of Meredith extensions preserve criticality and allow for the construction of graphs with controlled independence ratios?
• RQ4 Can the full conjecture be reduced to verifying it on a finite, ε-approximate family of graphs with bounded independence ratio?
• RQ5 What structural properties of k-critical graphs ensure that their independence ratio cannot exceed 1/2, and how can these be formalized via path systems?

Key findings

• For every ε > 0, there exists a family of critical graphs 𝒞_ε such that Vizing’s conjecture holds for all critical graphs if and only if it holds for all graphs in 𝒞_ε.
• Every graph in the constructed family 𝒞_ε has independence ratio strictly less than 1/2 + ε.
• The independence ratio of graphs in the class 𝒞(k,t) is bounded above by 1/2 + 1/(4kφ(k,t) + 2), where φ(k,t) = t(k−1)² + k−1.
• As t → ∞, the supremum of the independence ratio over 𝒞(k,t) converges to 1/2, proving the bound is tight in the limit.
• The result holds even when generalizing the number of distinguished paths per vertex from k−1 to any s ∈ {1,…,k−1}, with analogous convergence.
• The construction ensures that verifying the conjecture on graphs with independence ratio < 1/2 + ε is sufficient to prove it in full, reducing the problem to a finite, ε-approximate verification task.