## Abstract

In 1996 Kouider and Lonc proved the following natural generalization of Dirac’s Theorem: for any integer k ≥ 2, if G is an n-vertex graph with minimum degree at least n/k, then there are k - 1 cycles in G that together cover all the vertices. This is tight in the sense that there are n-vertex graphs that have minimum degree n/k − 1 and that do not contain k − 1 cycles with this property. A concrete example is given by I_{n,k} = K_{n}\K_{(k-1)n/k+1} (an edge-maximal graph on n vertices with an independent set of size (k - 1)n/k + 1). This graph has minimum degree n/k - 1 and cannot be covered with fewer than k cycles. More generally, given positive integers k1,…,k_{r} summing to k, the disjoint union I_{k1n/k,k1} + · ·· + I_{kr n/k,kr} is an n-vertex graph with the same properties. In this paper, we show that there are no extremal examples that differ substantially from the ones given by this construction. More precisely, we obtain the following stability result: if a graph G has n vertices and minimum degree nearly n/k, then it either contains k - 1 cycles covering all vertices, or else it must be close (in ‘edit distance’) to a subgraph of I_{k1n/k,k1} + · ·· + I_{kr n/k,kr}, for some sequence k_{1},…,k_{r} of positive integers that sum to k. Our proof uses Szemerédi’s Regularity Lemma and the related machinery.

Original language | English (US) |
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Article number | #P3.56 |

Journal | Electronic Journal of Combinatorics |

Volume | 24 |

Issue number | 3 |

DOIs | |

State | Published - Sep 8 2017 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics