@techreport{TR-IC-10-29,
   number = {IC-10-29},
   author = {C. N. Campos and S. Dantas and C. P. de Mello},
   title = {The total chromatic number of some families of snarks},
   month = {September},
   year = {2010},
   institution = {Institute of Computing, University of Campinas},
   note = {In English, 7 pages.
    \par\selectlanguage{english}\textbf{Abstract}
       The  \emph{total  chromatic  number}  $\chi_T(G)$  is the least
       number  of colours needed to colour the vertices and edges of a
       graph $G$, such that no incident or adjacent elements (vertices
       or edges) receive the same colour. It is known that the problem
       of  determining  the  total chromatic number is \np-hard and it
       remains \np-hard even for cubic bipartite graphs. \emph{Snarks}
       are  simple  connected  bridgeless  cubic  graphs which are not
       3-edge  colourable.  In  this  paper,  we  show  that the total
       chromatic  number  is  4 for three infinite families of snarks,
       namely,  the Flower Snarks, the Goldberg Snarks and the Twisted
       Goldberg Snarks. This result reinforces the conjecture that all
       snarks  are  type  1. Moreover, we give recursive procedures to
       construct 4-total colourings in each case.
  }
}