Strengthening Theorems of Dirac and Erdős on Disjoint Cycles

Henry Kierstead; A. V. Kostochka; A. Mcconvey

doi:10.1002/jgt.22106

Strengthening Theorems of Dirac and Erdős on Disjoint Cycles

Henry Kierstead, A. V. Kostochka, A. Mcconvey

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

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

Original language	English (US)
Pages (from-to)	788-802
Number of pages	15
Journal	Journal of Graph Theory
Volume	85
Issue number	4
DOIs	https://doi.org/10.1002/jgt.22106
State	Published - Aug 2017

Keywords

disjoint cycles
disjoint triangles
minimum degree
planar graphs

ASJC Scopus subject areas

Geometry and Topology

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10.1002/jgt.22106

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title = "Strengthening Theorems of Dirac and Erd{\H o}s on Disjoint Cycles",

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 Erd{\H o}s 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 Corr{\'a}di–Hajnal Theorem, which states that every graph G with Hk(G) = V(G) and |Hk(G)|≥3k contains k disjoint cycles.",

keywords = "disjoint cycles, disjoint triangles, minimum degree, planar graphs",

author = "Henry Kierstead and Kostochka, {A. V.} and A. Mcconvey",

note = "Funding Information: NSA; Contract grant number: H98230-12-1-0212; Contract grant sponsor: NSF; Contract grant numbers: DMS-1266016 and DMS-1600592; Contract grant sponsor: Russian Foundation for Basic Research; Contract grant numbers: 15-01-05867 and 16-01-00499; Contract grant sponsor: Campus Research Board, University of Illinois Publisher Copyright: {\textcopyright} 2016 Wiley Periodicals, Inc.",

year = "2017",

month = aug,

doi = "10.1002/jgt.22106",

language = "English (US)",

volume = "85",

pages = "788--802",

journal = "Journal of Graph Theory",

issn = "0364-9024",

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AU - Kierstead, Henry

AU - Kostochka, A. V.

AU - Mcconvey, A.

N1 - Funding Information: NSA; Contract grant number: H98230-12-1-0212; Contract grant sponsor: NSF; Contract grant numbers: DMS-1266016 and DMS-1600592; Contract grant sponsor: Russian Foundation for Basic Research; Contract grant numbers: 15-01-05867 and 16-01-00499; Contract grant sponsor: Campus Research Board, University of Illinois Publisher Copyright: © 2016 Wiley Periodicals, Inc.

PY - 2017/8

Y1 - 2017/8

N2 - 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.

AB - 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.

KW - disjoint cycles

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KW - minimum degree

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Strengthening Theorems of Dirac and Erdős on Disjoint Cycles

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