The color type of a spanning forest of a graph with colored edges is defined, and subsequently it is proved that the generating function of such spanning forests is obtained as the formal expansion of a certain determinant. An analogous determinantal expansion yields the generating function of all spanning forests of a given color type that contain a specific subforest.

Can a complete graph on an even number n (>4) of vertices be properly edge-colored with n-1 colors in such a way that the edges can be partitioned into edge disjoint colorful isomorphic spanning trees? A spanning treee is colorful if all n-1 colors occur among its edges. It is proved that this is possible to accomplish whenever n is a power of two.

Edge double ctitical graphs are defined and subsequently it is shown that the only edge double critical graph is the complete graph. Attention is then focussed on the Erd\"os-Lovasz vertex double critical conjecture. Some graph structure properties of a minimal counterexample are determined.

Select a set $S$ of 1-dimensional subspaces of a 2-dimensional vector $V$ space over $F_p$. Construct a graph whose vertex set is $V$ with vertices x and y adjacent if $x-y\in S.$ It is shown that two such graphs are isomorphic if and only if the underlying sets of 1-dimensional subspaces are in the same orbit of $PGL(2,p).$ This fact is used to construct infinite families of non-isomorphic self-complementary graphs of the same complexity. It answers certain complexity issues raised by Sedlacek. For all these graphs the chromatic number is shown to be equal to the size of the maximal clique. It is also shown that these graphs have maximal complexity among all graphs of the same size.