Improving bounds on the order of regular graphs of girth 5

Abstract

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