Color-critical graphs have logarithmic circumference

Asaf Shapira; Robin Thomas

doi:10.1016/j.aim.2011.05.001

Color-critical graphs have logarithmic circumference

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

Abstract

A graph G is k-critical if every proper subgraph of G is (k-1)-colorable, but the graph G itself is not. We prove that every k-critical graph on n vertices has a cycle of length at least logn/(100logk), improving a bound of Alon, Krivelevich and Seymour from 2000. Examples of Gallai from 1963 show that the bound cannot be improved to exceed 2(k-1)logn/log(k-2). We thus settle the problem of bounding the minimal circumference of k-critical graphs, raised by Dirac in 1952 and Kelly and Kelly in 1954.

Original language	English
Pages (from-to)	2309-2326
Number of pages	18
Journal	Advances in Mathematics
Volume	227
Issue number	6
DOIs	https://doi.org/10.1016/j.aim.2011.05.001
State	Published - 20 Aug 2011
Externally published	Yes

Keywords

Connectivity
Critical graphs
Long cycles

All Science Journal Classification (ASJC) codes

General Mathematics

Access to Document

10.1016/j.aim.2011.05.001

Cite this

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title = "Color-critical graphs have logarithmic circumference",

abstract = "A graph G is k-critical if every proper subgraph of G is (k-1)-colorable, but the graph G itself is not. We prove that every k-critical graph on n vertices has a cycle of length at least logn/(100logk), improving a bound of Alon, Krivelevich and Seymour from 2000. Examples of Gallai from 1963 show that the bound cannot be improved to exceed 2(k-1)logn/log(k-2). We thus settle the problem of bounding the minimal circumference of k-critical graphs, raised by Dirac in 1952 and Kelly and Kelly in 1954.",

keywords = "Connectivity, Critical graphs, Long cycles",

author = "Asaf Shapira and Robin Thomas",

note = "Funding Information: ✩ This material is based upon work supported by the National Science Foundation. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. * Corresponding author. E-mail address: [email protected] (A. Shapira). 1 Supported in part by NSF Grant DMS-0901355. 2 Supported in part by NSF Grant number DMS-0739366.",

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AU - Shapira, Asaf

AU - Thomas, Robin

N1 - Funding Information: ✩ This material is based upon work supported by the National Science Foundation. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. * Corresponding author. E-mail address: [email protected] (A. Shapira). 1 Supported in part by NSF Grant DMS-0901355. 2 Supported in part by NSF Grant number DMS-0739366.

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N2 - A graph G is k-critical if every proper subgraph of G is (k-1)-colorable, but the graph G itself is not. We prove that every k-critical graph on n vertices has a cycle of length at least logn/(100logk), improving a bound of Alon, Krivelevich and Seymour from 2000. Examples of Gallai from 1963 show that the bound cannot be improved to exceed 2(k-1)logn/log(k-2). We thus settle the problem of bounding the minimal circumference of k-critical graphs, raised by Dirac in 1952 and Kelly and Kelly in 1954.

AB - A graph G is k-critical if every proper subgraph of G is (k-1)-colorable, but the graph G itself is not. We prove that every k-critical graph on n vertices has a cycle of length at least logn/(100logk), improving a bound of Alon, Krivelevich and Seymour from 2000. Examples of Gallai from 1963 show that the bound cannot be improved to exceed 2(k-1)logn/log(k-2). We thus settle the problem of bounding the minimal circumference of k-critical graphs, raised by Dirac in 1952 and Kelly and Kelly in 1954.

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KW - Long cycles

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DO - 10.1016/j.aim.2011.05.001

M3 - مقالة

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SP - 2309

EP - 2326

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 6

ER -

Color-critical graphs have logarithmic circumference

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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