# Even orientations of graphs.

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Domenico Labbate

Dipartimento di Matematica, Informatica ed Economia - Università degli Studi della Basilicata - Potenza (Italy)

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Marien Abreu

Dipartimento di Matematica, Informatica ed Economia - Università degli Studi della Basilicata - Potenza (Italy)

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John Sheehan

Department of Mathematical Sciences, King's College, Aberdeen (Scotland)

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**Minisymposium:**
GENERAL SESSION TALKS

**Content:**
A graph $G$ is $1$--extendable if every edge belongs to at least one
$1$--factor.
%An orientation of a graph $G$ is an assignment of a {\em direction} to each
%edge of $G$. Now suppose that
Let $G$ be a graph with a $1$--factor $F$. Then an {\em even $F$--orientation} of
$G$ is an orientation in which each $F$--alternating cycle has exactly an even number
of edges directed in the same fixed direction around the cycle.
We examine the structure of $1$--extendible graphs $G$ which have no even
$F$--orientation where $F$ is a fixed $1$--factor of $G$ and we give a characterization for $k$--regular graphs with $k\ge 3$ and graphs with connectivity at least four.
Moreover, we will point out a
relationship between our results on even orientations and Pfaffian graphs.