Proof of the 1-factorization and Hamilton decomposition conjectures II: the bipartite case

Abstract

In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D\ge 2\lceil n/4\rceil -1$. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into perfect matchings. Equivalently, ${\chi }^{\prime }(G)=D$. (ii) [Hamilton decomposition conjecture] Suppose that $D\ge \lfloor n/2\rfloor$. 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 ${\mathrm{reg}}_{\mathrm{even}}(n,\delta )/2\ge (n-2)/8$ edge-disjoint Hamilton cycles. Here ${\mathrm{reg}}_{\mathrm{even}}(n,\delta )$ denotes the degree of the largest even-regular spanning subgraph one can guarantee in a graph on $n$ vertices with minimum degree $\delta$. According to Dirac, (i) was first raised in the 1950s. (ii) and the special case $\delta =\lceil n/2\rceil$ of (iii) answer questions of Nash-Williams from 1970. All of the above bounds are best possible. In the current paper, we prove the above results for the case when $G$ is close to a complete balanced bipartite graph.

We originally split the proof into four papers, of which this was the second paper. We have now combined this series into a single publication [arXiv:1401.4159v2], which will appear in the Memoirs of the AMS. 62 pages, 2 figures

References in corpus (2)

• Proof of the 1-factorization and Hamilton decomposition conjectures III: approximate decompositions
• Proof of the 1-factorization and Hamilton decomposition conjectures IV: exceptional systems for the two cliques case

Cited by in corpus (5)

• Hamilton cycles in graphs and hypergraphs: an extremal perspective
• Proof of the 1-factorization and Hamilton decomposition conjectures III: approximate decompositions
• Proof of the 1-factorization and Hamilton decomposition conjectures IV: exceptional systems for the two cliques case
• Packing, Counting and Covering Hamilton cycles in random directed graphs
• Vertex-transitive graphs that have no Hamilton decomposition