2025/06/24 by Araujo-Pardo, Gabriela, Mendoza-Cadena, Lydia Mirabel
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2506.20003
A [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]-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[(q)/(4),q;6]≤ 4q2-4, and if q is an odd prime power, and (q-3)/(2) is odd then our family satisfies that n[(q-1)/(4),q;6]≤ 4q2-4, otherwise n[(q-3)/(4),q;6]≤ 4q2-4.