Abstract
Let k ≥ 3 be an integer, Hk(G) be the set of vertices of degree at least 2k in a graph G, and Lk(G) be the set of vertices of degree at most 2k−2 in G. In 1963, Dirac and Erdős proved that G contains k (vertex) disjoint cycles whenever |Hk(G)|−|Lk(G)|≥k2+2k−4. The main result of this article is that for k≥2, every graph G with |V(G)|≥3k containing at most t disjoint triangles and with |Hk(G)|−|Lk(G)|≥2k + t contains k disjoint cycles. This yields that if |Hk(G)|−|Lk(G)|≥3k and k≥2, then G contains k disjoint cycles. This generalizes the Corrádi–Hajnal Theorem, which states that every graph G with Hk(G) = V(G) and |Hk(G)|≥3k contains k disjoint cycles.
Original language | English (US) |
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Pages (from-to) | 788-802 |
Number of pages | 15 |
Journal | Journal of Graph Theory |
Volume | 85 |
Issue number | 4 |
DOIs | |
State | Published - Aug 2017 |
Keywords
- disjoint cycles
- disjoint triangles
- minimum degree
- planar graphs
ASJC Scopus subject areas
- Geometry and Topology