New upper bounds on the order of mixed cages of girth 6

Abstract A $${[}z,r;g{]}$$ [ z , r ; g ] -mixed cage is a mixed graph of minimum order such that each vertex has z in-arcs, z out-arcs, r edges, and it has girth g , and the minimum order for $${[}z,r;g{]}$$ [ z , r ; g ] -mixed graphs is denoted by n [ z , r ; g ]. In this paper, we present an infinite family of mixed graphs with girth 6 that improves, in some cases, the families that we give in G. Araujo-Pardo and L. Mendoza-Cadena. On Mixed Cages of girth 6 , arXiv:2401.14768v2. In particular, if q is an even prime power, we construct a family of graphs that satisfies $$n[\frac{q}{4},q;6]\le 4q^{2}-4$$ n [ q 4 , q ; 6 ] ≤ 4 q 2 - 4 , and if q is an odd prime power, and $$\frac{q-3}{2}$$ q - 3 2 is odd, then our family satisfies that $$n[\frac{q-1}{4},q;6]\le 4q^2-4$$ n [ q - 1 4 , q ; 6 ] ≤ 4 q 2 - 4 , otherwise $$n[\frac{q-3}{4},q;6]\le 4q^2-4$$ n [ q - 3 4 , q ; 6 ] ≤ 4 q 2 - 4 .