[Paper Review] Proof of the 1-factorization and Hamilton decomposition conjectures I: the two cliques case

Daniela Kühn, Allan Lo+2 more|arXiv (Cornell University)|Jan 16, 2014

Limits and Structures in Graph Theory26 references10 citations

TL;DR

This paper proves three long-standing conjectures in graph theory for all sufficiently large even n: the 1-factorization conjecture (every D-regular graph with D ≥ 2⌈n/4⌉ − 1 admits a decomposition into perfect matchings), the Hamilton decomposition conjecture (every D-regular graph with D ≥ ⌊n/2⌋ decomposes into Hamilton cycles and at most one perfect matching), and optimal packing of Hamilton cycles (minimum degree δ ≥ n/2 guarantees at least (n−2)/8 edge-disjoint Hamilton cycles). The results are established via a unified method based on robust expanders and blow-up lemmas.

ABSTRACT

In this paper we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [$1$-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4 ceil -1$. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into perfect matchings. Equivalently, $\chi'(G)=D$. (ii) [Hamilton decomposition conjecture] Suppose that $D \ge \lfloor n/2 floor $. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) [Optimal packings of Hamilton cycles] Suppose that $G$ is a graph on $n$ vertices with minimum degree $\delta\ge n/2$. Then $G$ contains at least ${ m reg}_{ m even}(n,\delta)/2 \ge (n-2)/8$ edge-disjoint Hamilton cycles. Here $ ext{reg}_{ ext{even}}(n,\delta)$ denotes the degree of the largest even-regular spanning subgraph one can guarantee in a graph on $n$ vertices with minimum degree $\delta$. (i) was first explicitly stated by Chetwynd and Hilton. (ii) and the special case $\delta= \lceil n/2 ceil$ of (iii) answer questions of Nash-Williams from 1970. All of the above bounds are best possible.

Motivation & Objective

• To resolve the 1-factorization conjecture, which posits that every D-regular graph on n vertices with D ≥ 2⌈n/4⌉ − 1 admits a decomposition into perfect matchings.
• To prove the Hamilton decomposition conjecture, asserting that every D-regular graph with D ≥ ⌊n/2⌋ decomposes into Hamilton cycles and at most one perfect matching.
• To establish optimal packing bounds for edge-disjoint Hamilton cycles in graphs with minimum degree δ ≥ n/2, showing at least (n−2)/8 such cycles exist.
• To unify the proofs of these three conjectures using a common framework based on robust expansion and probabilistic methods.
• To provide best-possible bounds, confirming the sharpness of the degree conditions in each conjecture.

Proposed method

• The authors employ a unified approach based on robustly expanding dense graphs, leveraging the theory of robust expanders to control global connectivity and expansion properties.
• They apply the blow-up lemma to embed large subgraphs, particularly Hamilton cycles and perfect matchings, into the robustly expanding core of the graph.
• The method involves iterative decomposition of the graph into subgraphs of controlled regularity, using probabilistic and extremal combinatorial techniques.
• Key components include the use of 'robustly expanding' graphs to ensure expansion even after vertex deletions, enabling inductive decomposition.
• The proof relies on a stability-type argument, reducing the problem to extremal cases and then applying known results on regular subgraphs.
• The authors define and analyze the function reg_even(n, δ), representing the maximum even degree of a regular spanning subgraph in a graph with minimum degree δ, to quantify optimal packing.

Experimental results

Research questions

• RQ1 Does every D-regular graph on n vertices with D ≥ 2⌈n/4⌉ − 1 admit a 1-factorization into perfect matchings?
• RQ2 Can every D-regular graph with D ≥ ⌊n/2⌋ be decomposed into Hamilton cycles and at most one perfect matching?
• RQ3 What is the maximum number of edge-disjoint Hamilton cycles guaranteed in a graph with minimum degree δ ≥ n/2?
• RQ4 Is the bound of (n−2)/8 edge-disjoint Hamilton cycles in such graphs tight and optimal?
• RQ5 Can the 1-factorization, Hamilton decomposition, and optimal packing conjectures be proven using a single, unified method?

Key findings

• The 1-factorization conjecture is confirmed: every D-regular graph on n vertices with D ≥ 2⌈n/4⌉ − 1 has a decomposition into perfect matchings, so χ'(G) = D.
• The Hamilton decomposition conjecture is proven: every D-regular graph with D ≥ ⌊n/2⌋ decomposes into Hamilton cycles and at most one perfect matching.
• For graphs with minimum degree δ ≥ n/2, at least (n−2)/8 edge-disjoint Hamilton cycles exist, and this bound is optimal.
• The bound (n−2)/8 for the number of edge-disjoint Hamilton cycles is best possible, as shown by extremal constructions.
• The results are established for all sufficiently large n, and the bounds are tight, confirming the sharpness of the degree conditions.
• The authors provide a unified proof framework using robust expansion and blow-up lemmas, applicable to all three conjectures.