Improving bounds on the order of regular graphs of girth 5

Abstract

A (k,g)-graph is a k-regular graph with girth g and a (k,g)-cage is a (k,g)-graph with the fewest possible number of vertices n(k,g). Constructing (k,g)-cages and determining the order are both very hard problems. For this reason, an intensive line of research is devoted to constructing smaller (k,g)-graphs than previously known ones, providing in this way new upper bounds to n(k,g) each time such a graph is constructed.The paper focuses on girth g=5, where cages are known only for degrees k≤7. We construct (k,5)-graphs using and extending techniques of amalgamation into the incidence graphs of elliptic semiplanes of type L introduced and exposed by Funk (2009). The order of these graphs provides better upper bounds on n(k,5) than those known so far, for values of k such that either 13≤k≤33 or k≥66.