Maximal skew sets of lines on a Hermitian surface and a modified Bron-Kerbosch algorithm

TL;DR

This paper introduces a new algorithm for computing maximal skew sets of lines on Hermitian surfaces, providing computational results and bounds for various degrees, and addressing a variant of the clique listing problem.

Contribution

It presents a novel algorithm for finding maximal skew sets on Hermitian surfaces and solves a new variant of the clique listing problem.

Findings

Algorithm successfully computes skew sets for degrees 3, 4, and 5.

Constructs large skew sets on Hermitian varieties of any degree.

Provides lower bounds on sizes and counts of maximal skew sets.

Abstract

In this paper, we study maximal sets of skew lines on Hermitian surfaces. We give a new algorithm to compute these sets and give some computational results for Hermitian surfaces of degrees 3,4, and 5. In more generality, this algorithm solves a new variant of the clique listing problem, which may be more approachable than the classical problem. Finally, we explicitly construct a large skew set of lines on Hermitian varieties of any degree and use it to give a lower bound on the largest size of maximal skew sets and a lower bound on the possible number of maximal skew sets.