Orbits of toric promotion on bridge sums

TL;DR

The paper investigates toric promotion, a cyclic analogue of promotion on graphs, and its behavior under bridge-sum operations. Using coin and stone diagrams as a formalism, it proves that the orbit lengths for bridge sums of trees and complete graphs with simple graphs, as well as corona products with trees, are independent of the initial labeling on the non-bridge subgraphs and equal N(N−1) for the appropriate total vertex count N. It also provides explicit orbit-length formulas for bridge sums of complete graphs and for corona products, reinforcing a structural understanding of toric promotion under these graph constructions. These results extend Defant's earlier findings on trees and complete graphs and establish a unified view of orbit lengths under key graph operations, with proposed directions for iterative bridge sums and cycle interactions.

Abstract

In 2023, Defant introduced toric promotion as a cyclic analogue of Schützenberger's well known promotion operator. Toric promotion is defined by a choice of simple graph $G$ and acts on the labeling of $G$ by a series of involutions. Defant described the orbit length of toric promotion on trees and showed that it does not depend on the initial labeling; we prove an analogous result for complete graphs. A natural question is how toric promotion behaves under certain graph operations. In the main results of this article, we analyze the orbits of toric promotion under the bridge sum graph operation, which joins two graphs by adding an edge between a vertex of each graph. We show that the orbit length of toric promotion on any graph constructed via a bridge sum of a tree or a complete graph with a simple graph does not depend on the restriction of the initial labeling to the tree or complete subgraph. Additionally, we describe the orbit lengths of toric promotion on the bridge sums of two complete graphs and the bridge sums of a tree with a complete graph, and show that they do not depend on the initial labeling. Finally, we describe the orbit length of toric promotion on the corona product of a complete graph with any tree, and show that it does not depend on the initial labeling.

Figures (9)

• Figure 1: Let $G$ be the graph on $4$ vertices shown in yellow. We illustrate the orbit of toric promotion on $G$ containing the initial state $(4123,2)$. Each copy of $G$ shows a state of this orbit with the labels $i$ and $i+1$ shown in red.
• Figure 2: Let $G_1$ be the cycle graph on $3$ vertices and let $G_2$ be a tree on $4$ vertices. On the left, we show $G_1$ (yellow) and $G_2$ (blue). On the right, we show $G_1(a_1){ } G_2(b_4)$.
• Figure 3: Let $G_1$ be the cycle graph on $3$ vertices and let $G_2$ be a tree on $3$ vertices. On the left, we show $G_1$ (yellow) and $G_2$ (blue). On the right, we show $G_1\odot G_2(b_1)$.
• Figure 4: The stone diagram $\mathrm{SD}'$ (on the right) is a cyclic rotation of the stone diagram $\mathrm{SD}$ (on the left) by two positions clockwise, or $k=2$. In other words, $\mathrm{SD}'=\text{cyc}^2(\mathrm{SD})$.
• Figure 5: Let $G$ be the graph on $4$ vertices shown in yellow. We show the orbit of toric promotion on $G$ containing the initial state $(4123,2)$ in terms of stone and coin diagrams. Note that this is the same orbit as in \ref{['EX: toric promotion orbit']}.

Theorems & Definitions (27)

• Definition 1
• Theorem 2: D2023
• Proposition 3
• Definition 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Definition 9
• Theorem 10