@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.
}
}