Abstract
A graph G is 5/2-critical if G has no circular 5/2-coloring (or equivalently, homomorphism to C 5), but every proper subgraph of G has one. We prove that every 5/2-critical graph on n ≥ 4 vertices has at least
edges, and list all 5/2-critical graphs achieving this bound. This implies that every planar or projective-planar graph of girth at least 10 is 5/2-colorable.
Similar content being viewed by others
References
O. Borodin, S. G. Hartke, A. O. Ivanova, A. V. Kostochka and D. B. West: Circular (5,2)-coloring of sparse graphs, Sib. Elektron. Mat. Izv. 5 (2008), 417–426.
O. Borodin, S.-J. Kim, A. Kostochka and D. West: Homomorphisms from sparse graphs with large girth, Journal of Combinatorial Theory, Series B 90 (2004), 147–159.
M. DeVos: Open problem garden: Mapping planar graphs to odd cycles http://www.openproblemgarden.org/op/mapping_planar_graphs_to_odd_cycles.
H. Grötzsch: Ein Dreifarbensatz für Dreikreisfreie Netze auf der Kugel, Math.Natur. Reihe 8 (1959), 109–120.
F. Jaeger: On circular ows in graphs, in: Finite and Infinite Sets (Eger, 1981), Colloq. Math. Soc. J. Bolyai, vol. 37, 1984, 391–402.
W. Klostermeyer and C. Q. Zhang: (2+ε)-coloring of planar graphs with large odd-girth, J. Graph Theory 33 (2000), 109–119.
A. Kostochka and M. Yancey: Ore’s conjecture for k=4 and Grötzsch Theorem, Manuscript, 2012.
O. Ore: The Four Color Problem, Academic Press, New York, 1967.
A. Vince: Star chromatic number, Journal of Graph Theory 12 (1988), 551–559.
X. Zhu: Circular chromatic number: a survey, Discrete Mathematics 229 (2001), 371–410.
X. Zhu: Circular chromatic number of planar graphs of large odd girth, Electronic J. of Combinatorics 8 (2001), R25.
X. Zhu: Recent developments in circular colouring of graphs, in: Topics in discrete mathematics, Springer, 2006, 497–550.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by project GA14-19503S (Graph coloring and structure) of Czech Science Foundation.
Partially supported by NSERC under Discovery Grant No. 2014-06162, the Ontario Early Researcher Awards program and the Canada Research Chairs program.
Rights and permissions
About this article
Cite this article
Dvořák, Z., Postle, L. Density of 5/2-critical graphs. Combinatorica 37, 863–886 (2017). https://doi.org/10.1007/s00493-016-3356-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-016-3356-3