[Paper Review] Results on the intersection graphs of subspaces of a vector space

Nader Jafari Rad, Sayyed Heidar Jafari|arXiv (Cornell University)|May 4, 2011

Advanced Topics in Algebra2 references12 citations

TL;DR

This paper investigates the clique number, chromatic number, domination number, and independence number of the intersection graph of subspaces in a finite-dimensional vector space over a finite field. It establishes exact formulas for the clique and chromatic numbers when the dimension n is odd, and provides tight bounds for even n, while proving the domination number is q+1 and the independence number is (qⁿ−1)/(q−1).

ABSTRACT

For a vector space $V$ the \emph{intersection graph of subspaces} of $V$, denoted by $G(V)$, is the graph whose vertices are in a one-to-one correspondence with proper nontrivial subspaces of $V$ and two distinct vertices are adjacent if and only if the corresponding subspaces of $V$ have a nontrivial (nonzero) intersection. In this paper, we study the clique number, the chromatic number, the domination number and the independence number of the intersection graphs of subspaces of a vector space.

Motivation & Objective

• To characterize the clique number, chromatic number, domination number, and independence number of the intersection graph of subspaces in a vector space.
• To resolve the structure of intersection graphs for vector spaces over finite fields, particularly focusing on dimension-dependent behavior.
• To extend prior work on connectivity and graph-theoretic properties of such intersection graphs to fundamental invariants.
• To provide exact or tight bounds for key graph parameters, especially for even-dimensional spaces where the clique number remains an open problem.
• To unify algebraic and graph-theoretic techniques in the study of subspace intersection graphs.

Proposed method

• Define the intersection graph G(V) with vertices as proper nontrivial subspaces and edges when subspaces have nontrivial intersection.
• Use Gaussian binomial coefficients [n choose t]_q to count t-dimensional subspaces in an n-dimensional vector space over a finite field of size q.
• Apply Hall’s Marriage Theorem to construct perfect matchings in complement graphs of bipartite subgraphs between t- and (n−t)-dimensional subspaces.
• Use vertex coloring arguments based on matching structures to bound the chromatic number, leveraging Brooks’ Theorem and the clique number bound from Lemma 5.
• Construct dominating sets using quotient space subspaces of codimension 2 to prove the domination number is q+1.
• Establish the independence number via counting maximal partial spreads or using the fact that maximal independent sets correspond to sets of 1-dimensional subspaces covering the space.

Experimental results

Research questions

• RQ1 What is the exact value of the clique number w(G(V)) for vector spaces of even dimension?
• RQ2 How do the chromatic number and clique number of the intersection graph depend on the dimension and size of the underlying finite field?
• RQ3 What is the domination number γ(G(V)) in terms of the field size q and vector space dimension n?
• RQ4 What is the maximum size of an independent set in the intersection graph of subspaces?
• RQ5 How do the graph parameters behave differently for odd versus even-dimensional vector spaces?

Key findings

• For odd n, the clique number w(G(V)) and chromatic number χ(G(V)) are both equal to the sum of Gaussian binomial coefficients ∑_{i=1}^{⌊n/2⌋} [n choose i]_q.
• For even n, the clique number satisfies the lower bound ∑_{i=1}^{n/2−1} [n choose i]_q + [n−1 choose (n−2)/2]_q and the upper bound ∑_{i=1}^{n/2−1} [n choose i]_q + [n choose n/2]_q − q^{n²/4} − 1.
• The domination number γ(G(V)) is exactly q+1 for any n ≥ 2 and finite field of size q.
• The independence number α(G(V)) is exactly (qⁿ − 1)/(q − 1), corresponding to the number of 1-dimensional subspaces in V.
• The chromatic number for even n is bounded between the same expressions as the clique number, with the upper bound derived from degree and coloring arguments.
• The paper leaves open the exact value of the clique number for even-dimensional spaces, posing it as Problem 13.