Regular bipartite multigraphs have many (but not too many) symmetries

TL;DR

This work advances the understanding of symmetries in $(k,l)$-bipartite graphs by determining $\mu(k,l)$ and $\chi(k,l)$ for $k\ge 8$, revealing exact automorphism-group orders and showing the maximal case arises from disjoint black–white pairs. It connects automorphisms to $(k,l)$-intersection matrices (uniform contingency tables) and to uniform set partitions, establishing that in the typical, high-edge-count regime, automorphisms fix all vertices and have orders far smaller than the maximum. The paper also demonstrates, via probabilistic and combinatorial analysis, that random $(k,l)$-bipartite graphs have essentially trivial vertex automorphisms with a sharp upper bound $|\mathrm{Aut}(\Gamma)|\le l!^k/l^{(k-1)^2-\varepsilon}$ for any $\varepsilon>0$ and fixed $k$, while showing that the corresponding permutation groups on partitions are non-synchronizing. Together with the Baranyai-based construction in the synchronization section, these results contribute to a broader classification of automorphism behavior and synchronization phenomena in combinatorial structures.

Abstract

Let $k$ and $l$ be integers, both at least 2. A $(k,l)$-bipartite graph is an $l$-regular bipartite multigraph with coloured bipartite sets of size $k$. Define $χ(k,l)$ and $μ(k,l)$ to be the minimum and maximum order of automorphism groups of $(k,l)$-bipartite graphs, respectively. We determine $χ(k,l)$ and $μ(k,l)$ for $k\geq 8$, and analyse the generic situation when $k$ is fixed and $l$ is large. In particular, we show that almost all such graphs have automorphism groups which fix the vertices pointwise and have order far less than $μ(k,l)$. These graphs are intimately connected with both contingency tables with uniform margins and uniform set partitions; we examine the uniform distribution on the set of $k\times k$ contingency tables with uniform margin $l$, showing that with high probability all entries stray far from the mean. We also show that the symmetric group acting on uniform set partitions is non-synchronizing.

Theorems & Definitions (55)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3