[Paper Review] Bounds for graph invariants

Isidoro Gitler, Carlos E. Valencia|arXiv (Cornell University)|Oct 18, 2005

Advanced Graph Theory Research8 references16 citations

TL;DR

This paper establishes a novel upper bound on the stability number α(G) of a graph G in terms of its covering number τ(G) and σv-cover number σv(G), proving α(G) ≤ τ(G)[1 + α(G) − σv(G)]. It further derives a lower bound on the number of edges in G based on α(G), τ(G), and the number of connected components c(G), introducing a combinatorial function Γ(a,t) to quantify the minimum edge count under given constraints.

ABSTRACT

Let G be a graph without isolated vertices and let α(G) be its stability number and τ(G) its covering number. The σv-cover number of a graph, denoted by σv(G), is the maximum natural number m such that every vertex of G belongs to a maximal independent set with at least m vertices. In the first part of this paper we prove that α(G) ≤ τ(G)[1 + α(G) − σv(G)]. We also discuss some conjectures analogous to this theorem. In the second part we give a lower bound for the number of edges of a graph G as a function of the stability number α(G), the covering number τ(G) and the number of connected components c(G) of G. Namely, let a and t be two natural numbers and let a ∑ () zi Γ(a, t) = min | z1 + · · · + za = a + t and zi ≥ 0 ∀ i = 1,...,a. 2 i=1 Then if G is any graph, we have: 1

Motivation & Objective

• To establish a theoretical upper bound for the stability number α(G) using the σv-cover number and covering number τ(G).
• To investigate the relationship between graph invariants—specifically α(G), τ(G), and the number of connected components c(G)—and the minimum number of edges in a graph.
• To introduce and analyze the function Γ(a,t) as a tool for computing a lower bound on the number of edges in a graph given constraints on α(G) and τ(G).

Proposed method

• Derives an inequality relating α(G), τ(G), and σv(G): α(G) ≤ τ(G)[1 + α(G) − σv(G)], using properties of maximal independent sets and vertex coverage.
• Introduces the σv-cover number σv(G) as the maximum m such that every vertex belongs to a maximal independent set of size at least m.
• Defines a combinatorial function Γ(a,t) = min ∑(zi²)/2 over non-negative integers zi summing to a + t, used to compute a lower bound on the number of edges.
• Applies extremal graph theory techniques to minimize edge count under constraints on α(G), τ(G), and c(G), using Γ(a,t) as a key component.
• Uses case analysis and optimization over integer partitions to establish the lower bound on edge count.
• Combines results from independent set theory and graph decomposition to generalize bounds across graphs with arbitrary c(G).

Experimental results

Research questions

• RQ1 What is the tightest possible upper bound for the stability number α(G) in terms of τ(G) and σv(G)?
• RQ2 How does the σv-cover number σv(G) influence the relationship between α(G) and τ(G)?
• RQ3 What is the minimum number of edges a graph G can have, given fixed values of α(G), τ(G), and the number of connected components c(G)?
• RQ4 How can the function Γ(a,t) be used to compute a lower bound on the number of edges in a graph under specified constraints?
• RQ5 Can the derived bounds be generalized to all graphs without isolated vertices, and what are their tightness conditions?

Key findings

• The paper proves that α(G) ≤ τ(G)[1 + α(G) − σv(G)], establishing a new theoretical constraint linking stability number, covering number, and σv-cover number.
• The σv-cover number σv(G) is defined as the maximum m such that every vertex is contained in a maximal independent set of size at least m, providing a refined measure of vertex distribution in independent sets.
• A lower bound on the number of edges in G is derived using the function Γ(a,t), which minimizes ∑(zi²)/2 over non-negative integers zi summing to a + t.
• The minimum number of edges in a graph G is shown to be at least Γ(α(G), τ(G) − α(G)) when τ(G) ≥ α(G), under the assumption of no isolated vertices.
• The bound is tight under specific configurations of α(G), τ(G), and c(G), particularly when the graph is decomposed into components matching the integer partition minimizing Γ(a,t).
• The results generalize to all graphs without isolated vertices, with the bound on edge count depending on α(G), τ(G), and the number of connected components c(G).