[Paper Review] Tight lower bounds on the matching number in a graph with given maximum degree

Michael A. Henning, Anders Yeo|arXiv (Cornell University)|Apr 18, 2016

Advanced Graph Theory Research9 references3 citations

TL;DR

This paper establishes tight lower bounds on the matching number α′(G) in graphs with maximum degree at most k, proving that the set Lk of all pairs (γ, β) for which α′(G) ≥ γn + βm − K holds for all connected graphs G is a convex set. For odd k ≥ 3, Lk is the intersection of two closed half-spaces with one extreme point; for even k ≥ 4, it is the intersection of three half-spaces with two extreme points, fully characterizing the tightest possible linear bounds in terms of n (vertices) and m (edges).

ABSTRACT

Let $k \geq 3$. We prove the following three bounds for the matching number, $α'(G)$, of a graph, $G$, of order $n$ size $m$ and maximum degree at most $k$. If $k$ is odd, then $α'(G) \ge \left( \frac{k-1}{k(k^2 - 3)} ight) n \, + \, \left( \frac{k^2 - k - 2}{k(k^2 - 3)} ight) m \, - \, \frac{k-1}{k(k^2 - 3)}$. If $k$ is even, then $α'(G) \ge \frac{n}{k(k+1)} \, + \, \frac{m}{k+1} - \frac{1}{k}$. If $k$ is even, then $α'(G) \ge \left( \frac{k+2}{k^2+k+2} ight) m \, - \, \left( \frac{k-2}{k^2+k+2} ight) n \, - \frac{k+2}{k^2+k+2}$. In this paper we actually prove a slight strengthening of the above for which the bounds are tight for essentially all densities of graphs. The above three bounds are in fact powerful enough to give a complete description of the set $L_k$ of pairs $(γ,β)$ of real numbers with the following property. There exists a constant $K$ such that $α'(G) \geq γn + βm - K$ for every connected graph $G$ with maximum degree at most~$k$, where $n$ and $m$ denote the number of vertices and the number of edges, respectively, in $G$. We show that $L_k$ is a convex set. Further, if $k$ is odd, then $L_k$ is the intersection of two closed half-spaces, and there is exactly one extreme point of $L_k$, while if $k$ is even, then $L_k$ is the intersection of three closed half-spaces, and there are precisely two extreme points of $L_k$.

Motivation & Objective

• To determine the tightest possible linear lower bounds on the matching number α′(G) in terms of the number of vertices n and edges m for connected graphs with maximum degree at most k.
• To characterize the set Lk of all pairs (γ, β) for which α′(G) ≥ γn + βm − K holds for some constant K and all connected graphs G with Δ(G) ≤ k.
• To prove that Lk is a convex set and fully describe its geometric structure—specifically, its extreme points and defining half-spaces—for both odd and even k.
• To show that the derived bounds are tight for infinitely many graphs, including trees and k-regular graphs, across all possible edge densities.
• To unify and strengthen prior results on matching number lower bounds by providing a complete, tight, and exact characterization of the feasible region for (γ, β) pairs.

Proposed method

• Derives three tight lower bounds on α′(G) based on parity of k: one for odd k and two for even k, all expressed as linear functions of n and m with explicit constants.
• Uses the Tutte–Berge formula as a foundational tool to analyze the matching number via vertex subsets X that minimize (n + |X| − oc(G−X))/2.
• Introduces the concept of k-good and k-tight pairs (a,b) to classify which linear bounds (an + bm − K) are valid and tight for all graphs with maximum degree ≤ k.
• Applies geometric and convex analysis to show that Lk is the intersection of closed half-spaces defined by linear inequalities in (γ, β), with explicit coefficients derived from extremal graph families.
• Employs graph families such as trees and k-regular graphs, and a constructed family G′k,r, to prove tightness of bounds by constructing extremal examples.
• Uses interpolation and convex combination arguments (via Lemma 8) to show that all points in the convex hull of extreme points are also k-good, establishing the full convex structure of Lk.

Experimental results

Research questions

• RQ1 What is the tightest possible linear lower bound of the form α′(G) ≥ γn + βm − K that holds for all connected graphs G with maximum degree at most k?
• RQ2 How does the structure of the set Lk of valid (γ, β) pairs depend on the parity of k?
• RQ3 What are the extreme points of Lk, and how do they relate to extremal graph families such as trees and k-regular graphs?
• RQ4 Can the bounds be simultaneously tight for both sparse graphs (like trees) and dense graphs (like k-regular graphs), and for all intermediate densities?
• RQ5 What is the geometric nature of the set Lk, and how can it be fully described as an intersection of closed half-spaces?

Key findings

• For odd k ≥ 3, the set Lk is the intersection of two closed half-spaces ℓ₁ and ℓ₂, with exactly one extreme point: ( (k−1)/(k(k²−3)), (k²−k−2)/(k(k²−3)) ).
• For even k ≥ 4, Lk is the intersection of three closed half-spaces ℓ₁, ℓ₃, and ℓ₄, with exactly two extreme points: (1/(k(k+1)), 1/(k+1)) and (−(k−2)/(k²+k+2), (k+2)/(k²+k+2)).
• The bound for odd k is α′(G) ≥ ((k−1)/(k(k²−3)))n + ((k²−k−2)/(k(k²−3)))m − (k−1)/(k(k²−3)), and it is tight for infinitely many trees and k-regular graphs.
• For even k, two distinct bounds are derived: α′(G) ≥ n/(k(k+1)) + m/(k+1) − 1/k and α′(G) ≥ ((k+2)/(k²+k+2))m − ((k−2)/(k²+k+2))n − (k+2)/(k²+k+2), both tight for relevant graph families.
• The set Lk is convex, and all (γ, β) pairs in Lk yield valid linear lower bounds that are tight for infinitely many graphs across all edge densities.
• The paper proves that the derived bounds are not only valid but also optimal, as any pair (γ, β) outside Lk fails to bound α′(G) from below by γn + βm − K for all connected graphs with Δ(G) ≤ k.