A mixed cage is a finite mixed graph without loops or multiple edges, of minimum vertex
count for prescribed undirected degree 
, indegree and outdegree 
, and girth 
. The girth is the length of a shortest
cycle respecting the orientation of each directed graph
edge. Write the minimum order as 
.
De la Cruz and Pizaña (2026) proved that, for positive integers 
,
 |
They also proved
 |
when 
. These results concern mixed
graphs, not ordinary undirected cage graphs.
