Tight Bounds on the Vertex Ranking Number of Degenerate Graphs

Core Concepts

For any fixed integers d and ℓ, the ℓ-vertex-ranking number of n-vertex d-degenerate graphs is bounded by O(n^(1-2/(ℓ+1)) log n) for even ℓ and O(n^(1-2/ℓ) log n) for odd ℓ.

Abstract

The paper studies the ℓ-vertex-ranking number, χℓ-vr(G), of a graph G, which is the minimum number of colors needed to color the vertices of G such that in any connected subgraph H of G with diameter at most ℓ, there is a vertex in H whose color is larger than that of every other vertex in H.
The main results are:

For any fixed integers d and ℓ, the authors prove that every n-vertex d-degenerate graph G satisfies:

χℓ-vr(G) = O(n^(1-2/(ℓ+1)) log n) if ℓ is even
χℓ-vr(G) = O(n^(1-2/ℓ) log n) if ℓ is odd

This resolves (up to a logarithmic factor) an open problem posed by Karpas, Neiman, and Smorodinsky for the case ℓ = 2. The bounds are also asymptotically optimal (up to the log n factor) for ℓ ∈ {2, 3}.

The proofs rely on a theorem about graphs that are both d-degenerate and have maximum degree ∆. For such graphs, the authors show that χℓ-vr(G) = O(∆^(⌊ℓ/2⌋-1/2) log^(5/4) n), and if ∆^(⌊ℓ/2⌋-1) ≥ log n, then χℓ-vr(G) = O(∆^(⌊ℓ/2⌋-1/2) log n).

The key technical tools used in the proofs include an upper bound on the number of length-ℓ paths in a d-degenerate graph with maximum degree ∆, and a careful probabilistic analysis to bound the number of "problematic" paths that could lead to ℓ-violations in the vertex ranking.

Stats

The sum of vertex degrees in a d-degenerate n-vertex graph G is at most 2dn.
The number of vertices v in G with degree at least ∆ is at most 2dn/∆.

Quotes

"For any fixed d and ℓ, every n-vertex d-degenerate graph G satisfies χℓ-vr(G) = O(n^(1-2/(ℓ+1)) log n) if ℓ is even and χℓ-vr(G) = O(n^(1-2/ℓ) log n) if ℓ is odd."
"Theorem 2 exhibits a parity phenomenon one encounters when counting the number of paths of length ℓ in a d-degenerate graph of maximum-degree ∆."

Key Insights Distilled From

Vertex Ranking of Degenerate Graphs

by John Iacono,... at arxiv.org 04-26-2024

https://arxiv.org/pdf/2404.16340.pdf

Deeper Inquiries

How do the bounds in Theorem 2 compare to the best known lower bounds for the ℓ-vertex-ranking number of d-degenerate graphs

The bounds in Theorem 2 for the ℓ-vertex-ranking number of d-degenerate graphs provide significant improvements over the best-known lower bounds. The upper bounds in Theorem 2 are shown to be tight (up to a logn factor) for ℓ = 2 and ℓ = 3, matching the existing lower bounds for these cases. This is a significant advancement in the understanding of the vertex ranking of d-degenerate graphs, as it resolves open problems and establishes asymptotically optimal bounds for certain values of ℓ.

Can the logarithmic factors in the upper bounds of Theorem 2 be further improved

While the logarithmic factors in the upper bounds of Theorem 2 represent a substantial improvement in the understanding of vertex ranking in d-degenerate graphs, there is potential for further refinement. Future research could focus on reducing or optimizing these logarithmic factors to provide even tighter upper bounds. By exploring different techniques, algorithms, or mathematical approaches, it may be possible to achieve more precise upper bounds with smaller logarithmic terms, enhancing the accuracy and efficiency of vertex ranking in d-degenerate graphs.

What are the implications of these results for practical applications that involve vertex ranking, such as network analysis or resource allocation problems

The results of Theorem 2 have significant implications for practical applications involving vertex ranking, such as network analysis and resource allocation problems. By providing improved upper bounds on the ℓ-vertex-ranking number of d-degenerate graphs, these results offer more efficient and effective strategies for coloring vertices in various graph structures. This can lead to enhanced optimization in network design, improved resource allocation algorithms, and better decision-making processes in complex systems where vertex ranking plays a crucial role. The tighter bounds obtained can contribute to the development of more robust and scalable solutions for real-world problems in diverse fields such as telecommunications, computer networks, and social network analysis.