Existence of a self-complementary graph

Solution 1:

Hint:

Given a self complementary graph on $n$ vertices, try to construct a self-complementary graph on $n+4$ vertices.

Elaboration on the hint:

Given a self-complementary graph $G$ with $n$ vertices, add 4 more vertices which already form a 'Z' graph (the self-complementary graph for $n=4$). Connect the vertices of $G$ to some of four newly added vertices. Show that the new graph so formed, is a self-complementary graph on $n+4$ vertices.

Full answer:

Connect every vertex of $G$ to two of the vertices which are the farthest apart on the the n=4 graph (which is basically a path, and you connect all vertices of $G$ to the end points of that path).

This works even when $G$ has only one vertex.

Solution 2:

Recursive constructions are very popular. The construction given in this answer (which I posted in a previous life) seems to be much less popular, but it is also quite simple. To wit:

Take a complete graph with vertex set $V$ and edge set $E={V\choose2}$. Let $\alpha$ be any permutation of $V$ in which the length of each cycle is a multiple of $4$, except for at most one $1$-cycle. (Of course, such permutations exist if and only if $|V|\equiv 0$ or $1\pmod 4$.)

Let $\beta$ be the permutation of $E$ induced by $\alpha$. Observe that $\beta$ contains only cycles of even length. Color the edges in each cycle alternately black and white. The graph $G$ consisting of the black edges is self-complementary. (The permutation $\alpha$ is an anti-automorphism of $G.$)

Example. To construct self-complementary graphs of order $5$, take $V=\{a,b,c,d,e\}$ and let $\alpha=(a\;b\;c\;d)(e)$ so that $\beta=(ab\;bc\;cd\;ad)(ac\;bd)(ae\;be\;ce\;de)\;$. If we choose the edges $ab,cd$ and $ac$ (i.e. "color them black") we get a $4$-point path $P_4$. Now we can choose the edges $be,de$ obtaining the self-complementary graph $C_5$, or else we can choose $ae,ce$ obtaining the other self-complementary graph of order $5$, the one that looks like the letter A.

This construction is completely general, in that every self-complementary graph can be obtained as an instance of this construction.