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On Kotzig's conjecture in random graphs

Stefan Glock, Amedeo Sgueglia

TL;DR

This work studies Kotzig's perfect 1-factorisation conjecture through the lens of random graphs. It proves that for any fixed k, there exists a constant C(k) such that if p ≥ C(k) log n / n, the random graph G(n,p) w.h.p. contains k edge-disjoint perfect matchings whose pairwise unions are Hamiltonian cycles, effectively realizing a Kotzig-type structure in G(n,p). It also provides a precise counting result: for given edge-disjoint matchings M1,...,Mk in K_n, the probability that a uniformly random matching M* yields Hamilton cycles with all Mi is Θ_k(n^{-k/2})(n/e)^{n/2}. The proof combines the absorption method, random-process analysis, entropy arguments, and switching techniques, and leverages recent breakthroughs on the expectation threshold conjecture to transfer counting information to G(n,p). The paper further extends these ideas to the bipartite setting and discusses potential threshold refinements and connections to hitting-time phenomena in random graph processes. $ $

Abstract

In 1963, Anton Kotzig famously conjectured that $K_{n}$, the complete graph of order $n$, where $n$ is even, can be decomposed into $n-1$ perfect matchings such that every pair of these matchings forms a Hamilton cycle. The problem is still wide open and here we consider a variant of it for the binomial random graph $G(n,p)$. We prove that, for every fixed $k$, there exists a constant $C=C(k)$ such that, when $p\ge \frac{C \log n}{n}$, with high probability, $G(n,p)$ contains $k$ edge-disjoint perfect matchings with the property that every pair of them forms a Hamilton cycle. In fact, our main result is a very precise counting result for $K_n$. We show that, given any $k$ edge-disjoint perfect matchings $M_1,\dots,M_k$, the probability that a uniformly random perfect matching $M^*$ in $K_n$ has the property that $M^*\cup M_i$ forms a Hamilton cycle for each $i\in [k]$ is $Θ_k(n^{-k/2})$. This is proved by building on a variety of methods, including a random process analysis, the absorption method, the entropy method and the switching method. The result on the binomial random graph follows from a slight strengthening of our counting result via the recent breakthroughs on the expectation threshold conjecture.

On Kotzig's conjecture in random graphs

TL;DR

This work studies Kotzig's perfect 1-factorisation conjecture through the lens of random graphs. It proves that for any fixed k, there exists a constant C(k) such that if p ≥ C(k) log n / n, the random graph G(n,p) w.h.p. contains k edge-disjoint perfect matchings whose pairwise unions are Hamiltonian cycles, effectively realizing a Kotzig-type structure in G(n,p). It also provides a precise counting result: for given edge-disjoint matchings M1,...,Mk in K_n, the probability that a uniformly random matching M* yields Hamilton cycles with all Mi is Θ_k(n^{-k/2})(n/e)^{n/2}. The proof combines the absorption method, random-process analysis, entropy arguments, and switching techniques, and leverages recent breakthroughs on the expectation threshold conjecture to transfer counting information to G(n,p). The paper further extends these ideas to the bipartite setting and discusses potential threshold refinements and connections to hitting-time phenomena in random graph processes.

Abstract

In 1963, Anton Kotzig famously conjectured that , the complete graph of order , where is even, can be decomposed into perfect matchings such that every pair of these matchings forms a Hamilton cycle. The problem is still wide open and here we consider a variant of it for the binomial random graph . We prove that, for every fixed , there exists a constant such that, when , with high probability, contains edge-disjoint perfect matchings with the property that every pair of them forms a Hamilton cycle. In fact, our main result is a very precise counting result for . We show that, given any edge-disjoint perfect matchings , the probability that a uniformly random perfect matching in has the property that forms a Hamilton cycle for each is . This is proved by building on a variety of methods, including a random process analysis, the absorption method, the entropy method and the switching method. The result on the binomial random graph follows from a slight strengthening of our counting result via the recent breakthroughs on the expectation threshold conjecture.

Paper Structure

This paper contains 18 sections, 24 theorems, 49 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Notation, organisation and basic concepts
  4. Notation
  5. Organisation of the paper
  6. Basic concepts
  7. Existence -- Proof of Theorem \ref{['thm:abs']}
  8. Existence of absorbers
  9. Finding absorbers inside configurations.
  10. Proof of \ref{['thm:abs']}
  11. Counting -- Proof of Theorem \ref{['thm:count']}
  12. General observations
  13. Lower bound
  14. Upper bound
  15. Switching lemma
  16. ...and 3 more sections

Key Result

Theorem 1.2

For every fixed $k \in \mathbb{N}$ there exists a constant $C=C(k)$ such that, when $p\ge \frac{C \log n}{n}$, w.h.p. $G(n,p)$ contains $k$ edge-disjoint perfect matchings $M_1, \dots, M_k$ such that $M_i \cup M_j$ induces a Hamilton cycle for all distinct $i,j \in [k]$.

Figures (5)

  • Figure 1: A perfect $1$-factorisation of $K_6$, where each $1$-factor is drawn with a different colour.
  • Figure 2: (A): A $3$-configuration $\mathcal{M}$ of order $8$ and a partial red matching $M$ of size $2$. (B): The reduced configuration of $\mathcal{M}$ with respect to $M$ (note that each colour still induces a perfect matching and that we have created double coloured edges). (C): A red perfect matching $M'$ of the reduced configuration. (D): $M \cup M'$ is a red perfect matching of $\mathcal{M}$.
  • Figure 3: (A): A $3$-configuration $\mathcal{L}$ on $10$ vertices $\{v_i\}$ and its blue-green equalizer $\mathcal{A}$ with vertices $\{x_i\} \cup \{y_i\}$: The empty matching is blue-green equalizing for $\mathcal{A}$ and $M_1 \cup M_2$ is blue-green equalizing for $\mathcal{A} \cup \mathcal{L}$. (B): We show how we add the blue and green matching on $Z$ and how we choose the red edges in $M_1$ and $M_2$ in the proof of \ref{['lem:equalizer']}. (B1): We first add double blue-green coloured edges $x_iy_i$ and set $M_1:=\{v_ix_i\}$. (B2): The blue and green edges of the reduced configuration of $M_1$. (B3): We find a red matching $M_2$ which equalizes blue and green. (B4): Indeed the blue and green edges of the reduced configuration of $M_1 \cup M_2$ are identical. (C): We show how we add the yellow matching on $Z$ in the proof of \ref{['lem:equalizer']}. (C1): Once $M_1$ and $M_2$ have been fixed, we arbitrarily choose two distinct vertices $y, \bar{y}$ not covered by a red edge. (C2): We can greedily add a yellow matching covering all but $4$ vertices such that $y, \bar{y}$ are not covered and we do not close a red-yellow cycle. (C3): We add the yellow edges $sy, \bar{s}\bar{y}$ which completes a yellow perfect matching.
  • Figure 4: Let $e=ab$. Then $e$ is blocked in colour green, but not blocked in colour blue.
  • Figure 5: Switching between $B^i_\ell$ and $N^i_{\ell-1}$ in \ref{['lem:B_ell_N_ell-1']}.

Theorems & Definitions (33)

  • Conjecture 1.1: Kotzig's conjecture kotzig:64
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: $k$-configuration
  • Definition 2.2: Reduced configuration
  • Definition 2.3: Overlap
  • Definition 2.4: Good and bad edges
  • Proposition 2.5
  • Proposition 3.1
  • ...and 23 more