Strengthening Theorems of Dirac and Erdős on Disjoint Cycles

Abstract

Let k≥3 be an integer, Hkfalse(Gfalse) be the set of vertices of degree at least 2k in a graph G, and Lkfalse(Gfalse) 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 false|Hk(G)false|−false|Lk(G)false|≥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 false|Hk(G)false|−false|Lk(G)false|≥2k+t contains k disjoint cycles. This yields that if k≥2 and false|Hk(G)false|−false|Lk(G)false|≥3k, then G contains k disjoint cycles. This generalizes the Corrádi–Hajnal Theorem, which states that every graph G with Hkfalse(Gfalse)=Vfalse(Gfalse) and false|Hk(G)false|≥3k contains k disjoint cycles.

A Sharp Dirac–Erdős Type Bound for Large Graphs

“…As shown in [12], if a graph G with |G| 3k and δ (G) 2k − 1 does not contain a large independent set, then with two exceptions, G contains k disjoint cycles. This corollary, along with the following theorem from [10] will be used in the proof. We prove the following technical statement that implies Theorem 1.7, but is more amenable to induction.…”

“…Indeed, K 3k−1 contains at most k − 1 disjoint cycles, so for small graphs, a bound of at least 3k is necessary. The authors [10] recently proved that 3k is also sufficient.…”

“…In [10], the authors describe another graph G 1 (k), obtained from G 0 (k) by adding k vertices of degree 1, each adjacent to x. The graph G 1 (k) still contains only k − 1 disjoint cycles, but has 4k vertices and |H k (G 1 (k))| − |L k (G 1 (k))| = 2k.…”

“…The graph G 1 (k) still contains only k − 1 disjoint cycles, but has 4k vertices and |H k (G 1 (k))| − |L k (G 1 (k))| = 2k. However, in the special case that G is planar, it is shown in [10] that the bound of 2k is sufficient.…”

“…Further, the graph G 1 (k) described above shows that a difference of 2k is not sufficient when |G| is small. In [10], we were not able to determine whether for each k there are only finitely many such examples. In order to attract attention to this problem and based on known examples, we raised the following question.…”