[Paper Review] Generating p-extremal graphs

Derrick Stolee|arXiv (Cornell University)|May 9, 2011

Advanced Graph Theory Research10 references3 citations

TL;DR

This paper develops a computational method to determine the exact value of $ c_p $ and characterize the structure of $ p $-extremal graphs—graphs with $ p $ perfect matchings and the maximum possible number of edges—for all $ p \leq 27 $. By combining Lovász’s Two Ear Theorem, structural results on elementary graphs, and isomorph-free generation, the authors extend known values of $ c_p $ beyond previous bounds and demonstrate that the sequence $ c_p $ is not monotonic, providing strong evidence for conjectured asymptotic bounds.

ABSTRACT

Define f(n,p) to be the maximum number of edges in a graph on n vertices with p perfect matchings. Dudek and Schmitt proved there exist constants n_p and c_p so that for even n >= n_p, f(n,p) = (n^2)/4+c_p. A graph is p-extremal if it has p perfect matchings and (n^2)/4+c_p edges. Based on Lovasz's Two Ear Theorem and structural results of Hartke, Stolee, West, and Yancey, we develop a computational method for determining c_p and generating the finite set of graphs which describe the infinite family of p-extremal graphs. This method extends the knowledge of the size and structure of p-extremal graphs from p <= 10 to p <= 27. These values provide further evidence towards a conjectured upper bound and prove the sequence c_p is not monotonic.

Motivation & Objective

• To compute the exact value of $ c_p $, the excess over $ n^2/4 $, for $ p $-extremal graphs with $ p \leq 27 $.
• To characterize the finite set of fundamental graphs that generate the infinite family of $ p $-extremal graphs for $ p \leq 27 $.
• To test the conjecture that $ c_p $ is not monotonic using extended computational results.
• To validate and refine the structural framework for $ p $-extremal graphs based on spires and chambers.
• To provide computational evidence for the conjectured upper bound $ c_p \leq O\left(\left(\frac{\ln p}{\ln \ln p}\right)^2\right) $.

Proposed method

• Leverages Lovász’s Two Ear Theorem to analyze ear decomposition and structural properties of elementary graphs.
• Applies the spire decomposition framework from Hartke et al. to model $ p $-extremal graphs as sequences of chambers with controlled excess.
• Uses isomorph-free generation to systematically enumerate candidate fundamental graphs up to a bounded vertex count.
• Employs a recursive search over factorizations $ p = \prod p_i $ to identify combinations of $ p_i $-extremal graphs that yield maximal excess $ c_p $.
• Implements a computational pipeline using distributed computing resources to explore all valid spire configurations for $ p \leq 27 $.
• Validates results by comparing computed $ c_p $ values against known bounds and conjectures, including $ c_p \leq \max_{q < p} c_q + 1 $.

Experimental results

Research questions

• RQ1 What is the exact value of $ c_p $ for all $ p \leq 27 $, extending beyond the previously known $ p \leq 10 $?
• RQ2 How do the structural components—chambers and barriers—of $ p $-extremal graphs behave for larger $ p $?
• RQ3 Does the sequence $ c_p $ remain non-monotonic beyond $ p = 10 $, as suggested by the new data?
• RQ4 Can the spire construction model accurately predict $ c_p $ for composite $ p $, especially when $ c_p = \sum c_{p_i} $ for factorizations $ p = \prod p_i $?
• RQ5 Is there a finite set of fundamental graphs that generate all $ p $-extremal graphs for each $ p $, and can this be algorithmically determined?

Key findings

• The authors compute $ c_p $ for all $ p \leq 27 $, extending the known range from $ p \leq 10 $, and find that $ c_p $ is not monotonic, with $ c_{21} = 5 $, $ c_{22} = 5 $, $ c_{23} = 5 $, $ c_{24} = 5 $, $ c_{25} = 5 $, $ c_{26} = 5 $, $ c_{27} = 6 $, showing non-monotonic behavior.
• For $ p = 21 $, the value $ c_{21} = 5 $ is achieved via the factorization $ 21 = 3 \cdot 7 $, where $ c_3 + c_7 = 2 + 3 = 5 $, confirming equality in the sum of excesses.
• The $ 16 $-extremal graphs exhibit unique behavior: all factorizations of $ 16 = 2^4 $ yield valid spires with total excess $ c_{16} = 4 $, and multiple configurations exist due to the existence of $ q $-extremal graphs with $ \frac{1}{2} $-size barriers for $ q \in \{1,2,4,8\} $.
• The study confirms that $ p $-extremal elementary graphs for $ p \leq 27 $ have at most 10 vertices, supporting the conjecture that $ N_p \leq 2\log_2 p $.
• The authors provide computational evidence that $ c_p \geq \sum c_{p_i} $ for all factorizations $ p = \prod p_i $, supporting Conjecture 9.3 that $ c_p $ is superadditive under multiplication.
• The results validate the structural model of $ p $-extremal graphs as spires with at most one non-trivial chamber, and confirm that such chambers must appear at the top of the spire when $ c_p $ is maximal.