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An Algorithm for the Decomposition of Complete Graph into Minimum Number of Edge-disjoint Trees

Antika Sinha, Sanjoy Kumar Saha, Partha Basuchowdhuri

TL;DR

The paper tackles decomposing the edge set of a complete graph $K_n$ into the minimum number of edge-disjoint trees. Using Tutte and Nash-Williams results and the tree covering number $\tau$, arboricity $\alpha$, and spanning tree packing number $\sigma$, it proves the optimal count is $\tau(K_n)=\lceil \frac{n}{2} \rceil$ with a corresponding size pattern. It then introduces two linear-time constructions, DECK-E for even $n$ and DECK-O for odd $n$, that realize the decomposition in $O(m)$ time. Experiments on $K_6$ and $K_7$ illustrate the method and show strong alignment with known bounds for even orders, with the constructed decompositions offering practical benefits for load balancing, parallel computation, and network design.

Abstract

In this work, we study methodical decomposition of an undirected, unweighted complete graph ($K_n$ of order $n$, size $m$) into minimum number of edge-disjoint trees. We find that $x$, a positive integer, is minimum and $x=\lceil\frac{n}{2}\rceil$ as the edge set of $K_n$ is decomposed into edge-disjoint trees of size sequence $M = \{m_1,m_2,...,m_x\}$ where $m_i\le(n-1)$ and $Σ_{i=1}^{x} m_i$ = $\frac{n(n-1)}{2}$. For decomposing the edge set of $K_n$ into minimum number of edge-disjoint trees, our proposed algorithm takes total $O(m)$ time.

An Algorithm for the Decomposition of Complete Graph into Minimum Number of Edge-disjoint Trees

TL;DR

The paper tackles decomposing the edge set of a complete graph into the minimum number of edge-disjoint trees. Using Tutte and Nash-Williams results and the tree covering number , arboricity , and spanning tree packing number , it proves the optimal count is with a corresponding size pattern. It then introduces two linear-time constructions, DECK-E for even and DECK-O for odd , that realize the decomposition in time. Experiments on and illustrate the method and show strong alignment with known bounds for even orders, with the constructed decompositions offering practical benefits for load balancing, parallel computation, and network design.

Abstract

In this work, we study methodical decomposition of an undirected, unweighted complete graph ( of order , size ) into minimum number of edge-disjoint trees. We find that , a positive integer, is minimum and as the edge set of is decomposed into edge-disjoint trees of size sequence where and = . For decomposing the edge set of into minimum number of edge-disjoint trees, our proposed algorithm takes total time.
Paper Structure (6 sections, 5 theorems, 6 equations, 6 figures, 3 tables, 2 algorithms)

This paper contains 6 sections, 5 theorems, 6 equations, 6 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Problem Statement
  4. Algorithm
  5. Results
  6. Discussion

Key Result

Theorem 1

(Tutte tutte1961problem & Nash-Williams nash1961edge, 1961): Let $G$ be a graph and $\kappa$ be a positive integer. Then $G$ contains $\kappa$ edge-disjoint spanning trees if and only if holds for every partition $P$ of $V(G)$ and $E_P(G)$ is the set of the edges of $G$ joining vertices belonging to different members of $P$. Therefore, $G_P$ denotes a graph of vertex set P and edge set $E_P$(G).

Figures (6)

  • Figure 1: The resulting tree set $T[1:3]$ obtained using algorithm \ref{['algo:decke']}: DECK-E where the input graph is $K_6$.
  • Figure 2: Decomposition of an even-ordered complete graph (DECK-E($K_n$)) into the minimum number of edge-disjoint trees
  • Figure 3: Decomposition of an odd-ordered complete graph (DECK-O($K_n$)) into the minimum number of edge-disjoint trees
  • Figure 4: The resulting tree set $T[1:4]$ obtained using algorithm \ref{['algo:decko']}: DECK-O where the input graph is $K_7$.
  • Figure 5: The tree set $T[1:7]$ obtained by decomposing edge set of $K_{n=14}$ into the minimum number ($\tau=7$) of edge-disjoint trees, following our proposed Algorithm \ref{['algo:decke']} DECK-E, designed specifically for even values of $n$.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Lemma 2
  • Lemma 3