Skip to main content

A Sharp Dirac–Erdős Type Bound for Large Graphs

  • H. A. KIERSTEAD (a1), A. V. KOSTOCHKA (a2) (a3) and A. McCONVEY (a4)

Let k ⩾ 3 be an integer, hk(G) be the number of vertices of degree at least 2k in a graph G, and ℓk(G) be the number of vertices of degree at most 2k − 2 in G. Dirac and Erdős proved in 1963 that if hk(G) − ℓk(G) ⩾ k2 + 2k − 4, then G contains k vertex-disjoint cycles. For each k ⩾ 2, they also showed an infinite sequence of graphs Gk(n) with hk(Gk(n)) − ℓk(Gk(n)) = 2k − 1 such that Gk(n) does not have k disjoint cycles. Recently, the authors proved that, for k ⩾ 2, a bound of 3k is sufficient to guarantee the existence of k disjoint cycles, and presented for every k a graph G0(k) with hk(G0(k)) − ℓk(G0(k)) = 3k − 1 and no k disjoint cycles. The goal of this paper is to refine and sharpen this result. We show that the Dirac–Erdős construction is optimal in the sense that for every k ⩾ 2, there are only finitely many graphs G with hk(G) − ℓk(G) ⩾ 2k but no k disjoint cycles. In particular, every graph G with |V(G)| ⩾ 19k and hk(G) − ℓk(G) ⩾ 2k contains k disjoint cycles.

Hide All
[1] Chiba, S., Fujita, S., Kawarabayashi, K.-I. and Sakuma, T. (2014) Minimum degree conditions for vertex-disjoint even cycles in large graphs. Adv. Appl. Math. 54 105120.
[2] Corrádi, K. and Hajnal, A. (1963) On the maximal number of independent circuits in a graph. Acta Math. Hungar. 14 423439.
[3] Dirac, G. (1963) Some results concerning the structure of graphs. Canad. Math. Bull. 6 183210.
[4] Dirac, G. and Erdős, P. (1963) On the maximal number of independent circuits in a graph. Acta Math. Hungar. 14 7994.
[5] Enomoto, H. (1998) On the existence of disjoint cycles in a graph. Combinatorica 18 487492.
[6] Erdős, P. (1989) On some aspects of my work with Gabriel Dirac. Ann. Discrete Math. 41 111116.
[7] Hajnal, A. and Szemerédi, E. (1970) Proof of a conjecture of P. Erdős. In Combinatorial Theory and its Applications II (Proc. Colloq., Balatonfüred, 1969), North-Holland, pp. 601–623.
[8] Kierstead, H. and Kostochka, A. (2008) An Ore-type theorem on equitable coloring. J. Combin. Theory Ser. B 98 226234.
[9] Kierstead, H. and Kostochka, A. (2015) A refinement of a result of Corrádi and Hajnal. Combinatorica 35 497512.
[10] Kierstead, H., Kostochka, A. and McConvey, A. (2017) Strengthening theorems of Dirac and Erdős on disjoint cycles. J. Graph Theory 85 788802.
[11] Kierstead, H., Kostochka, A., Molla, T. and Yeager, E. (2017) Sharpening an Ore-type version of the Corrádi–Hajnal theorem. Abh. Math. Semin. Univ. Hambg. 87 299335.
[12] Kierstead, H., Kostochka, A. and Yeager, E. (2017) On the Corrádi–Hajnal theorem and a question of Dirac. J. Combin. Theory Ser. B 122 121148.
[13] Wang, H. (1999) On the maximum number of independent cycles in a graph. Discrete Math. 205 183190.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed