@techreport{TR-IC-13-18,
   number = {IC-13-18},
   author = {At\'{i}lio Gomes Luiz and R. Bruce Richter},
   title = {{Colorings and crossings}},
   month = {August},
   year = {2013},
   institution = {Institute of Computing, University of Campinas},
   note = {In English, 32 pages.
    \par\selectlanguage{english}\textbf{Abstract}
       In  2007,  Albertson  conjectured  that  if  a  graph  $G$  has 
       chromatic  number  $k$,  then  the crossing number of $G$ is at
       least  the  crossing  number  of  the  complete  graph with $k$
       vertices.  To  date,  two  papers  were written on this subject
       trying  to  solve the conjecture for an arbitrary $k$-chromatic
       graph $G$, and after much effort the conjecture was proved true
       for  $k  \leq  16$.  In  this report, we present an overview of
       topics related to Albertson's Conjecture, such as: the crossing
       number  problem,  lower bounds on crossing number, lower bounds
       on  the  number of edges of color-critical graphs, and coloring
       of  graphs  with small crossing number and small clique number.
       While investigating Albertson's conjecture, J. Bar\'{a}t and G.
       T\'{o}th    proposed   the   following   conjecture   involving  
       color-critical  graphs:  for  every positive integer $c$, there
       exists  a  bound  $k(c)$  such  that for any $k$, where $k \geq
       k(c)$, any k-critical graph on $k+c$ vertices has a subdivision
       of  $K_k$.  In  Section $9$, we present counterexamples to this
       conjecture  for  every  $c  \geq  6$  and  we  prove  that  the 
       conjecture is valid for $c = 5$.
  }
}