[Paper Review] Problems and results in Extremal Combinatorics -- IV

Noga Alon|arXiv (Cornell University)|Sep 26, 2020

Limits and Structures in Graph Theory28 references4 citations

TL;DR

This paper presents new results in extremal combinatorics, focusing on graph packing with high girth, coding theory, and social choice. It establishes a sufficient condition for packing two graphs such that their union maintains a high girth, generalizing classical results and enabling explicit constructions of high-girth directed expanders.

ABSTRACT

Extremal Combinatorics is among the most active topics in Discrete Mathematics, dealing with problems that are often motivated by questions in other areas, including Theoretical Computer Science and Information Theory. This paper contains a collection of problems and results in the area, including solutions or partial solutions to open problems suggested by various researchers. The topics considered here include questions in Extremal Graph Theory, Coding Theory and Social Choice. This is by no means a comprehensive survey of the area, and is merely a collection of problems, results and proofs, which are hopefully interesting. As the title of the paper suggests, this is a sequel of three previous papers of the same flavour. Each section of this paper is essentially self contained, and can be read separately.

Motivation & Objective

• Investigate conditions under which two graphs can be packed such that their union maintains a high girth, extending classical graph packing theorems.
• Generalize the Erdős–Sachs result on the existence of high-girth regular graphs by incorporating degree and girth constraints in packing.
• Provide an explicit construction method for high-girth directed expanders using the new packing condition.
• Address open problems in dominating set theory, particularly the gap between connected and standard domination numbers in regular graphs.
• Explore the interplay between graph structure, girth, and degree constraints in extremal combinatorics.

Proposed method

• Use a potential function argument based on conditional expectations to iteratively construct a dominating set with bounded size.
• Define a conditional expectation function $ \psi_i $ that combines the expected size of the dominating set, the number of undominated vertices, and a concave function of the sum of inverse degrees.
• Apply linearity of expectation to compute conditional expectations for each vertex in the sequence, ensuring monotonic decrease of the potential function.
• Use the concavity of the function $ f $ to ensure that choosing the next vertex to minimize $ \psi_{i+1} $ leads to a decreasing potential, guaranteeing a good approximation.
• Prove that the final value of the potential function is bounded by $ \frac{n}{k+1} (\ln(k+1) + 4) - 2 $, leading to a bound on the size of the dominating set.
• Apply the method to analyze the difference between connected and standard domination numbers, showing it is at most $ \frac{n}{k+1} (\ln \lceil \ln(k+1) \rceil + 3) $.

Experimental results

Research questions

• RQ1 Under what conditions can two graphs with given maximum degrees and girth be packed so that their union has high girth?
• RQ2 Can the Bollobás–Eldridge conjecture on graph packing be extended to preserve high girth in the combined graph?
• RQ3 What is the maximum possible difference between the connected domination number and the standard domination number in $ k $-regular connected graphs?
• RQ4 How can high-girth directed expanders be explicitly constructed using extremal graph packing results?
• RQ5 What is the tightest possible upper bound on the domination gap $ \gamma_c(G) - \gamma(G) $ for $ k $-regular connected graphs?

Key findings

• The paper proves that if $ (d_1 + d_2) $-ball of radius $ k $ in a graph has size less than $ n $, then two graphs with maximum degrees $ d_1, d_2 $ and girth at least $ g $ can be packed so that the combined graph has girth at least $ \min\{g, k\} $.
• For fixed $ d_1 + d_2 \geq 3 $, the value of $ k $ satisfying the size condition is asymptotically $ (1+o(1)) \frac{\log n}{\log(d_1 + d_2 - 1)} $, which controls the girth of the packed graph.
• The result implies the existence of high-girth regular graphs, recovering the Erdős–Sachs theorem as a special case.
• An explicit construction of high-girth directed expanders is derived from the packing result.
• The difference between connected and standard domination numbers in $ k $-regular connected graphs is at most $ \frac{n}{k+1} (\ln \lceil \ln(k+1) \rceil + 3) $, and at least $ \lfloor \frac{n}{k+1} \rfloor - 1 $, leaving a $ \ln \ln(k+1) $ gap to close.
• The algorithm based on conditional expectation yields a dominating set whose size is bounded by $ \frac{n}{k+1} (\ln(k+1) + 4) - 2 $, improving approximation guarantees in regular graphs.