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{\bf Ilse Fischer and C.H.C. Little }
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{\bf Even Circuits of Prescribed Clockwise Parity}
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We show that a graph has an orientation under which every circuit
of even length is clockwise odd if and only if the graph contains
no subgraph which is, after the contraction of at most one circuit
of odd length, an even subdivision of $K_{2,3}$. In fact we give a
more general characterisation of graphs that have an orientation
under which every even circuit has a prescribed clockwise parity.
Moreover we show that this characterisation has an equivalent analogue
for signed graphs.

We were motivated to study the original problem by our work on
Pfaffian graphs, which are the graphs that have an orientation under
which every alternating circuit is clockwise odd. Their significance
is that they are precisely the graphs to which Kasteleyn's powerful
method for enumerating perfect matchings may be applied.

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