  A \emph{self-complementary} graph is a graph isomorphic to its complement. An
isomorphism between $G$ and its complement, viewed as a permutation of $V(G)$,
is then called an \emph{antimorphism}. A \emph{skew partition} of $G$ is a
partition of $V(G)$ into 4 sets $A,B,C,D$ such that there is no edge between
$A,B$ and every possible edge between $C,D$. A \emph{symmetric partition} of
$G$ is a partition of $V(G)$ into 4 sets $A,B,C,D$ such that there is no edge
between $A, D$, no edge between $B, C$, every possible edge between $A,B$ and
every possible edge between $C,D$.
  We give a new proof of a theorem of Gibbs saying that every
self-complementary graph on $4k$ vertices has $k$ disjoint paths on 4 vertices
as induced subgraph. This new proof gives more structural information than the
original one. We conjecture that every self-complementary graph on $4k$
vertices either has an induced cycle on 5 vertices, or a skew partition, or a
symmetric partition. The new proof of Gibb's theorem yields a proof of the
conjecture for the self-complementary graphs that have an antimorphism that is
the product of a two circular permutations, one of them of length 4.
