Abstract
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) ⩾ k 2 + 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 G 0(k) with hk (G 0(k)) − ℓk(G 0(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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1-11 |
| Number of pages | 11 |
| Journal | Combinatorics Probability and Computing |
| DOIs | |
| State | Accepted/In press - Mar 9 2018 |
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ASJC Scopus subject areas
- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics
Cite this
A Sharp Dirac–Erdős Type Bound for Large Graphs. / Kierstead, Henry; KOSTOCHKA, A. V.; McCONVEY, A.
In: Combinatorics Probability and Computing, 09.03.2018, p. 1-11.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - A Sharp Dirac–Erdős Type Bound for Large Graphs
AU - Kierstead, Henry
AU - KOSTOCHKA, A. V.
AU - McCONVEY, A.
PY - 2018/3/9
Y1 - 2018/3/9
N2 - 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) ⩾ k 2 + 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 G 0(k) with hk (G 0(k)) − ℓk(G 0(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.
AB - 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) ⩾ k 2 + 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 G 0(k) with hk (G 0(k)) − ℓk(G 0(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.
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UR - http://www.scopus.com/inward/citedby.url?scp=85043307486&partnerID=8YFLogxK
U2 - 10.1017/S0963548318000020
DO - 10.1017/S0963548318000020
M3 - Article
AN - SCOPUS:85043307486
SP - 1
EP - 11
JO - Combinatorics Probability and Computing
JF - Combinatorics Probability and Computing
SN - 0963-5483
ER -