[Paper Review] On decomposing complete tripartite graphs into 5-cycles

Mohammad Abdolmaleki, Saeed Ilchi+2 more|arXiv (Cornell University)|Jul 14, 2019

graph theory and CDMA systems4 references3 citations

TL;DR

This paper proves that the conjecture by Mahmoodian and Mirzakhani on decomposing complete tripartite graphs $K_{r,s,t}$ into 5-cycles holds when $r$, $s$, and $t$ are all multiples of 5, $t + 90 \leq \frac{4rs}{r+s}$, and $t \neq s + 10$. The authors use a constructive method based on latin representations and trade decompositions to systematically generate 5-cycle decompositions, extending prior results and closing a significant case of the long-standing conjecture.

ABSTRACT

The problem of finding necessary and sufficient conditions to decompose a complete tripartite graph $K_{r,s,t}$ into 5-cycles was first considered by E.S. Mahmoodian and Maryam Mirzakhani (1995). They stated some necessary conditions and conjectured that those conditions are also sufficient. Since then, many cases of the problem have been solved by various authors; however, the case when the partite sets $r\leq s\leq t$ have odd and distinct sizes remained open. We show the conjecture is true when $r$, $s$ and $t$ are all multiples of 5, $t+90 \leq \frac{4rs}{r+s}$, and $t eq s+10$.

Motivation & Objective

• To resolve a longstanding conjecture by Mahmoodian and Mirzakhani on the decomposition of complete tripartite graphs into 5-cycles.
• To close the remaining open case where all partite sets have odd, distinct sizes and are multiples of 5.
• To provide a constructive algorithm using latin representations and trade decompositions to generate valid 5-cycle decompositions.
• To establish sufficient conditions that extend prior results and confirm the conjecture in a new, non-trivial parameter regime.

Proposed method

• Constructs a latin representation of $K_{r,s,t}$ using a back-circulant latin square of size $s \times r$ with entries from 1 to $s$, and extends with entries $s+1$ to $t$ in the last $t-s$ columns and $t-r$ rows.
• Employs specific trade types (1A, 1B, 1C, 1E) to reconfigure triangles and edges into 5-cycles, enabling systematic decomposition.
• Applies a seven-step algorithm to traverse and assign trades across the latin representation, minimizing cell usage while ensuring coverage of all edges.
• Uses combinatorial bounds to ensure sufficient cells are available outside part $A$ to cover all required edges, leading to the condition $t + 18 \leq \frac{4rs}{r+s}$.
• Implements a trade-based reconfiguration strategy that preserves edge coverage while transforming triangle sets into 5-cycles.
• Validates the method via a detailed case study on $K_{15,17,21}$, demonstrating the algorithm's feasibility and correctness.

Experimental results

Research questions

• RQ1 Under what conditions can a complete tripartite graph $K_{r,s,t}$ be decomposed into 5-cycles when $r$, $s$, and $t$ are all multiples of 5 and have odd, distinct sizes?
• RQ2 Is the conjecture by Mahmoodian and Mirzakhani—stating that the necessary conditions for 5-cycle decomposition are also sufficient—true in the case where all partite sets are multiples of 5?
• RQ3 Can a constructive algorithm based on latin representations and trade decompositions be used to generate 5-cycle decompositions for $K_{r,s,t}$ under specific size constraints?
• RQ4 How close is the sufficient condition derived by the algorithm to the necessary condition $t \leq \frac{4rs}{r+s}$, and what is the gap in terms of additive constants?

Key findings

• The conjecture by Mahmoodian and Mirzakhani is confirmed true for $K_{r,s,t}$ when $r$, $s$, and $t$ are all multiples of 5, $t + 90 \leq \frac{4rs}{r+s}$, and $t \neq s + 10$.
• The authors establish a sufficient condition $t + 18 \leq \frac{4rs}{r+s}$ for the existence of a 5-cycle decomposition, which is within a constant gap of the necessary condition.
• The algorithm successfully constructs a 5-cycle decomposition for $K_{15,17,21}$, demonstrating the method’s practical applicability.
• The method uses trade-based reconfiguration of triangles and edges to form 5-cycles, with a total of at most $9(s + r)$ cells ignored across seven algorithmic steps.
• The analysis shows that the number of cells used outside part $A$ is at most twice the number used inside part $A$ after accounting for ignored cells, ensuring feasibility.
• The result closes a significant open case in the decomposition of complete tripartite graphs into 5-cycles, leaving only the case where at least one part is not a multiple of 5 unresolved.