2014/01/16 by Csaba, Béla, Kühn, Daniela, Lo, Allan +2
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1401.4178
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≥ 2\lceil n/4\rceil -1. Then every D-regular graph G on n vertices has a decomposition into perfect matchings. Equivalently, χ'(G)=D. (ii) [Hamilton decomposition conjecture] Suppose that D ≥ \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) We prove an optimal result on the number of edge-disjoint Hamilton cycles in a graph of given minimum degree. According to Dirac, (i) was first raised in the 1950s. (ii) and (iii) answer questions of Nash-Williams from 1970. The above bounds are best possible. In the current paper, we show the following: suppose that G is close to a complete balanced bipartite graph or to the union of two cliques of equal size. If we are given a suitable set of path systems which cover a set of `exceptional' vertices and edges of G, then we can extend these path systems into an approximate decomposition of G into Hamilton cycles (or perfect matchings if appropriate).