2020/09/29 by Araujo-Pardo, Gabriela, de la Cruz, Claudia, González-Moreno, Diego
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2009.13709
A [z, r; g]-mixed cage is a mixed graph z-regular by arcs, r-regular by edges, with girth g and minimum order. %In this paper we study structural properties of mixed cages: Let n[z,r;g] denote the order of a [z,r;g]-mixed cage. In this paper we prove that n[z,r;g] is a monotonicity function, with respect of g, for z∈ \1,2\, and we use it to prove that the underlying graph of a [z,r;g]-mixed cage is 2-connected, for z∈ \1,2\. We also prove that [z,r;g]-mixed cages are strong connected. We present bounds of n[z,r;g] and constructions of [z,r;5]-mixed graphs and show a [10,3;5]-mixed cage of order 50.