It seems that many, if not almost all, of the properties studied in graph theory are monotone. (*Property* means it is invariant under permutation of vertices, and *monotone* means that the property is either preserved under addition or deletion of edges, fixing the vertex set.) For example: connected, planar, triangle-free, bipartite, etc. Many quantitative graph invariants can also be considered monotone graph properties, e.g. chromatic number $\ge k$ or girth $\ge g$.

My question is whether there are non-monotone graph properties which are well studied, or which arise naturally.

An obvious class of examples is the intersection of a monotone increasing and monotone decreasing property: for example graphs with chromatic number $\ge k$ *and* girth $\ge g$. (It is not entirely obvious if you intersect two such properties that they will have a nonempty intersection -- in this case it is a well-known theorem in graph theory.

Another example is the presence of induced subgraphs isomorphic to $H$ for any graph $H$. Adding edges only increases the number of subgraphs, but it can destroy the property of being induced.

I am especially interested to hear if any non-monotone properties have been studied for random graphs. A famous theorem of Friedgut and Kalai is that every monotone graph property has a sharp threshold, and I would like to know about any examples of sharp thresholds for non-monotone properties.