  The local chromatic number of a graph was introduced by Erd\H{o}s et al. [4].
In [17] a connection to topological properties of (a box complex of) the graph
was established and in [18] it was shown that if a graph is strongly
topologically 4-chromatic then its local chromatic number is at least four. As
a consequence one obtains a generalization of the following theorem of Youngs:
If a quadrangulation of the projective plane is not bipartite it has chromatic
number four. The generalization states that in this case the local chromatic
number is also four. Both papers [1] and [13] generalize Youngs's result to
arbitrary non-orientable surfaces replacing the condition of the graph being
not bipartite by a more technical condition of an odd quadrangulation. This
paper investigates when these general results are true for the local chromatic
number instead of the chromatic number. Surprisingly, we find out that (unlike
in the case of the chromatic number) this depends on the genus of the surface.
For the non-orientable surfaces of genus at most four, the local chromatic
number of any odd quadrangulation is at least four, but this is not true for
non-orientable surfaces of genus 5 or higher. We also prove that face
subdivisions of odd quadrangulations and Fisk triangulations of arbitrary
surfaces exhibit the same behavior for the local chromatic number as they do
for the usual chromatic number.
