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Sub-exponentially many 3-colorings of triangle-free planar graphs [article]

Arash Asadi, Zdenek Dvorak, Luke Postle, Robin Thomas
2011 arXiv   pre-print
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

Arash Asadi, Luke Postle, Robin Thomas
2009 Electronic Notes in Discrete Mathematics  
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]

Benjamin Moore, Evelyne Smith-Roberge
2022 arXiv   pre-print
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