TY - JOUR
T1 - Color-critical graphs have logarithmic circumference
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.
PY - 2011/8/20
Y1 - 2011/8/20
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.
KW - Connectivity
KW - Critical graphs
KW - Long cycles
UR - https://www.scopus.com/pages/publications/79957630872
U2 - 10.1016/j.aim.2011.05.001
DO - 10.1016/j.aim.2011.05.001
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AN - SCOPUS:79957630872
SN - 0001-8708
VL - 227
SP - 2309
EP - 2326
JO - Advances in Mathematics
JF - Advances in Mathematics
IS - 6
ER -