I know that nauty can generate n-vertex bipartite graphs (!geng n -c -b), but I haven't seen a corresponding command for its opposite (non-bipartite graphs). (I might have missed it.)
SageMath provides a way to filter graphs as follows:
s=[g for g in graphs.nauty_geng('-c 6') if g.is_bipartite()==False]
len(s)
95
But I would like to ask if there is a closer approach to nauty, such as using a C version to obtain non-bipartite connected graphs. That's because nauty is implemented in C. Alternatively, is there a direct algorithm for generating non-bipartite graphs?
When I asked Brendan McKay, he told me this can works.
pickg -~b
Besides, pickg can do many things.
Constraints:
Numerical constraints (shown here with following #) can take a single integer
value, or a range like #:#, #:, or :#. Each can also be preceded by '~', which
negates it. (For example, -~D2:4 will match any maximum degree which is _not_ 2,
3, or 4.) Constraints are applied to all input graphs, and only those which match
all constraints are counted or selected.
-n# number of vertices -e# number of edges
-d# minimum degree -D# maximum degree
-m# vertices of min degree -M# vertices of max degree
-r regular -b bipartite
-z# radius -Z# diameter
-g# girth (0=acyclic) -Y# total number of cycles
-T# number of triangles -K# number of maximal independent sets
-H# number of induced cycles
-E Eulerian (all degrees are even, connectivity not required)
-a# group size -o# orbits -F# fixed points -t vertex-transitive
-c# connectivity (only implemented for 0,1,2).
-i# min common nbrs of adjacent vertices; -I# maximum
-j# min common nbrs of non-adjacent vertices; -J# maximum