An odd chordless cycle, also called an odd hole, is a chordless cycle whose length is an odd number greater than 4.
By König's line coloring theorem, the line graph of a bipartite graph is a perfect graph and hence has no odd chordless cycles. It can nevertheless contain triangles and therefore need not itself be a bipartite graph.
A graph is a perfect graph iff neither the graph 
nor its graph complement 
has an odd
chordless cycle. A graph with no 5-cycle
graph and no larger odd chordless cycle
is therefore automatically a perfect graph. This
is true since the presence of a chordless 5-cycle graph
in 
corresponds to a 5-cycle graph in 
and 
can have no chordless 7-cycle
graph or larger since the diagonals of these cycle
graphs in 
would contain a 5-cycle graph in 
.
