Small bi-regular graphs of even girth

Abstract

A graph of girth g that contains vertices of degrees r and m is called a bi-regular ({r,m},g)-graph. As with the Cage Problem, we seek the smallest ({r,m},g)-graphs for given parameters 2≤r<m, g≥3, called ({r,m},g)-cages. The orders of the majority of ({r,m},g)-cages, in cases where m is much larger than r and the girth g is odd, have been recently determined via the construction of an infinite family of graphs whose orders match a well-known lower bound, but a generalization of this result to bi-regular cages of even girth proved elusive.We summarize and improve some of the previously established lower bounds for the orders of bi-regular cages of even girth and present a generalization of the odd girth construction to even girths that provides us with a new general upper bound on the order of graphs with girths of the form g=2t, t odd. This construction produces infinitely many ({r,m};6)-cages with sufficiently large m. We also determine a ({3,4};10)-cage of order 82.