Generating pivot Gray codes for spanning trees of complete graphs in constant amortized time

Abstract

We present the first known pivot Gray code for spanning trees of complete graphs, listing all spanning trees such that consecutive trees differ by pivoting a single edge around a vertex. This pivot Gray code thus addresses an open problem posed by Knuth in The Art of Computer Programming, Volume 4 (Exercise 101, Section 7.2.1.6, [Knuth, 2011]), rated at a difficulty level of 46 out of 50, and imposes stricter conditions than existing revolving-door or edge-exchange Gray codes for spanning trees of complete graphs. Our recursive algorithm generates each spanning tree in constant amortized time using $O(n^2)$ space. In addition, we provide a novel proof of Cayley's formula, $n^{n-2}$, for the number of spanning trees in a complete graph, derived from our recursive approach. We extend the algorithm to generate edge-exchange Gray codes for general graphs with $n$ vertices, achieving $O(n^2)$ time per tree using $O(n^2)$ space. For specific graph classes, the algorithm can be optimized to generate edge-exchange Gray codes for spanning trees in constant amortized time per tree for complete bipartite graphs, $O(n)$-amortized time per tree for fan graphs, and $O(n)$-amortized time per tree for wheel graphs, all using $O(n^2)$ space.

Accepted by SODA 2026