Mixed Cages : monotony, connectivity and upper bounds

Abstract

A [z, r; g]-mixed cage is a mixed graph z-regular by arcs, r-regular by edges, with girth g and minimum order. 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.