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2-factors in Dense Graphs Eldar Fischer

Summary: 2-factors in Dense Graphs
Noga Alon
Eldar Fischer
A conjecture of Sauer and Spencer states that any graph G on n vertices with minimum degree at
least 2
3 n contains any graph H on n vertices with maximum degree 2 or less. This conjecture is proven
here for all sufficiently large n.
1 Introduction
All graphs considered here are finite and have no loops and no parallel edges. A 2-factor of a graph G is
a 2-regular spanning subgraph of G, that is, a spanning subgraph every connected component of which
is a cycle. In the following discussion Ck will always denote a cycle of k vertices.
CorrŽadi and Hajnal [8] proved that any graph G on at least 3k vertices with minimum degree at least
2k contains k vertex disjoint cycles. In particular, a graph on n = 3k vertices with minimum degree at
least 2
3 n contains a 2-factor consisting of k vertex disjoint triangles. It is easy to see that this is tight, as
the complete 3-partite graph with vertex classes of sizes k - 1, k and k + 1 has minimum degree 2
3 n - 1
and does not contain k vertex disjoint triangles. The problem of determining the best possible minimum
degree of a graph G that ensures it contains a 2-factor of a prescribed type has been considered by various


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics