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Sub-exponentially many 3-colorings of triangle-free planar graphs
[article]
2011
arXiv
pre-print
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Thomassen conjectured that every triangle-free planar graph on n vertices has exponentially many 3-colorings, and proved that it has at least 2^[n^(1/12)/20000] distinct 3-colorings. ...
We show that it has at least 2^sqrt(n/362) distinct 3-colorings. ...
The theorem does not extend verbatim to any non-planar surface, but Thomassen proved that every graph of girth at least five embedded in the projective plane or the torus is 3-colorable. ...
arXiv:1007.1430v2
fatcat:ljahmr6smrc4tc4bbribvtqksa
Sub-Exponentially Many 3-Colorings of Triangle-Free Planar Graphs
2009
Electronic Notes in Discrete Mathematics
Preserved Fulltext
A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2013; you can also visit the original URL.
The file type is application/pdf.
Thomassen conjectured that every triangle-free planar graph on n vertices has exponentially many 3-colorings, and proved that it has at least 2 n 1/12 /20000 distinct 3-colorings. ...
We show that it has at least 2 √ n/362 distinct 3-colorings. ...
The theorem does not extend verbatim to any non-planar surface, but Thomassen proved that every graph of girth at least five embedded in the projective plane or the torus is 3-colorable. ...
doi:10.1016/j.endm.2009.07.014
fatcat:opdqk3sqvvhxrnz6yysxak2yrq
A density bound for triangle-free 4-critical graphs
[article]
2022
arXiv
pre-print
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A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2020; you can also visit the original URL.
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2020 pre-print arXiv:2012.01503v1 fatcat:griyk3vkdnf6fjgjmknlaqc4qm
We show every triangle-free 4-critical graph G satisfies e(G) ≥5v(G)+2/3. In fact, we prove a more general result characterizing sparse 4-critical graphs with few vertex- disjoint triangles. ...
Every graph of girth at least five embeddable on the torus or the projective plane is 3-colourable.
. ...
In Section 4, we give a brief overview of the potential method and results specific to its use for colour-critical graphs. ...
arXiv:2012.01503v2
fatcat:hvew363qzzap5nqf4jbodmpe6u