Improved length lower bounds for reflecting sequences

Abstract

We present a computational method for finding lower bounds on the lengths of reflecting sequences for labeled chains. The method, called Quadtree Enumeration, is similar to an existing algorithm for finding such bounds, but exhibits several improvements and optimizations, and hence runs much faster. Using the new approach, we have shown a length lower bound of 19t – 214 for t-reflecting sequences for labeled 7-chains, thus improving the current length lower bound for universal traversal sequences for 2-regular graphs of n vertices from Ω(n1.48) to Ω(n1.51), and for universal traversal sequences for d-regular graphs of n vertices, where \(3 \leqslant d \leqslant \frac{n}{{17}} + 1\), from Ω(d2−1.48n2.48) to Ω(d2−1.51n2.51).