  We give an elementary proof for the non-$3$-colorability of $4$-chromatic
Schrijver graphs thus providing such a proof also for $4$-chromatic Kneser
graphs. To this end we use a complete description of the structure of
$4$-chromatic Schrijver graphs that was already given by Braun and even earlier
in an unpublished manuscript by Li. We also address connections to surface
quadrangulations. In particular, we show that a spanning subgraph of
$4$-chromatic Schrijver graphs quadrangulates the Klein bottle, while another
spanning subgraph quadrangulates the projective plane. The latter is a special
case of a result by Kaiser and Stehl\'{\i}k. We characterize the color-critical
edges of $4$-chromatic Schrijver graphs and also present preliminary results
toward the characterization of color-critical edges in Schrijver graphs of
higher chromatic number. Finally, we show that (apart from two cases of small
parameters) the subgraphs we present that quadrangulate the Klein bottle are
edge-color-critical. The analogous result for the subgraphs quadrangulating the
projective plane is an immediate consequence of earlier results by Gimbel and
Thomassen and was already noted by Kaiser and Stehl\'{\i}k.
