[Paper Review] Partitions of graphs into small and large sets

Asen Bojilov, Yair Caro+2 more|RECERCAT (Consorci de Serveis Universitaris de Catalunya)|May 8, 2012

Advanced Graph Theory Research7 references4 citations

TL;DR

This paper introduces k-small and k-large sets in graphs, defining partition parameters φₖ(G) and Ωₖ(G) for minimum partitions into such sets. It establishes tight connections to independence number, clique number, and chromatic number, develops linear-time greedy algorithms for φₖ(G) and Ωₖ(G), and derives sharp inequalities that refine the Caro-Wei, Turán, and Hansen-Zheng theorems via new bounds on edge count and degree sequences.

ABSTRACT

Let $G$ be a graph on $n$ vertices. We call a subset $A$ of the vertex set $V(G)$ \emph{$k$-small} if, for every vertex $v \in A$, $°(v) \le n - |A| + k$. A subset $B \subseteq V(G)$ is called \emph{$k$-large} if, for every vertex $u \in B$, $°(u) \ge |B| - k - 1$. Moreover, we denote by $φ_k(G)$ the minimum integer $t$ such that there is a partition of $V(G)$ into $t$ $k$-small sets, and by $Ω_k(G)$ the minimum integer $t$ such that there is a partition of $V(G)$ into $t$ $k$-large sets. In this paper, we will show tight connections between $k$-small sets, respectively $k$-large sets, and the $k$-independence number, the clique number and the chromatic number of a graph. We shall develop greedy algorithms to compute in linear time both $φ_k(G)$ and $Ω_k(G)$ and prove various sharp inequalities concerning these parameters, which we will use to obtain refinements of the Caro-Wei Theorem, the Turán Theorem and the Hansen-Zheng Theorem among other things.

Motivation & Objective

• To define and analyze k-small and k-large sets in graphs as generalizations of independent and clique sets.
• To develop efficient greedy algorithms for computing φₖ(G) and Ωₖ(G), the minimum partition sizes into k-small and k-large sets.
• To establish tight connections between φₖ(G), Ωₖ(G), and classical graph parameters like independence number, clique number, and chromatic number.
• To derive sharp inequalities involving φₖ(G) and Ωₖ(G) that refine known theorems such as Caro-Wei, Turán, and Hansen-Zheng.
• To provide new upper and lower bounds on the number of edges in a graph using φₖ(G) and Ωₖ(G), improving upon classical results.

Proposed method

• Define a k-small set A as a subset where every vertex v ∈ A satisfies deg(v) ≤ n − |A| + k, and a k-large set B where deg(v) ≥ |B| − k − 1 for all v ∈ B.
• Introduce φₖ(G) as the minimum number of k-small sets into which V(G) can be partitioned, and Ωₖ(G) analogously for k-large sets.
• Establish duality via complementation: φₖ(G) = Ωₖ(Ḡ) and Ωₖ(G) = φₖ(Ḡ), linking the parameters across a graph and its complement.
• Develop greedy algorithms that compute φₖ(G) and Ωₖ(G) in O(n + m) time by iteratively selecting maximal k-small or k-large sets.
• Derive bounds on edge count e(G) using φₖ(G) and Ωₖ(G), showing e(G) ≤ (φₖ(G) − 1)n² / (2φₖ(G)) and similar inequalities.
• Introduce α- and β-variants of small/large sets (e.g., α-small: ∑₁/(deg(v)+1) ≤ 1) and define φₖ^α(G), Ωₖ^α(G), etc., to refine bounds.

Experimental results

Research questions

• RQ1 How do k-small and k-large sets relate to classical graph parameters such as independence number, clique number, and chromatic number?
• RQ2 Can efficient algorithms be designed to compute φₖ(G) and Ωₖ(G) in linear time for any graph G?
• RQ3 What are the tightest possible inequalities involving φₖ(G) and Ωₖ(G), and how do they refine known theorems like Caro-Wei and Turán?
• RQ4 To what extent do the α- and β-variants of small and large sets improve bounds on edge count and degree sequences?
• RQ5 Is the inequality φₖ^β(G) ≤ CW(Ḡ) always true, and how does it relate to the Caro-Wei bound?

Key findings

• φₖ(G) and Ωₖ(G) can be computed in linear time O(n + m) using greedy algorithms that iteratively select maximal k-small or k-large sets.
• The parameters satisfy φₖ(G) = Ωₖ(Ḡ) and Ωₖ(G) = φₖ(Ḡ), establishing a strong duality between a graph and its complement.
• The inequality φₖ(G) ≥ n / (n − d(G)) holds, and equality is approached in specific constructions, such as when G is a join of two cliques.
• The bound e(G) ≤ (φₖ(G) − 1)n² / (2φₖ(G)) improves upon Turán’s classical bound when φₖ(G) < ω(G), providing tighter edge count estimates.
• For the Caro-Wei bound, the inequality CW(Ḡ) ≤ φₖ^α(G) ≤ φₖ^β(G) ≤ φₖ(G) holds, and φₖ^β(G) ≤ CW(Ḡ) is conjectured but not proven.
• The maximum size of a k-small set Sₖ(G) satisfies Sₖ(G) ≤ Sₖ^α(G) ≤ Sₖ^β(G) ≤ ⌊(n−Δ)/2 + √((n−Δ)²/4 + nΔ − 2e(G))⌋, providing a closed-form upper bound.