On girth-biregular graphs

Abstract

Let Γ denote a finite, connected, simple graph. For an edge e of Γ let n(e) denote the number of girth cycles containing e. For a vertex v of Γ let {e 1 , e 2 , . . . , e k } be the set of edges incident to v ordered such that n(e 1 ) ≤ n(e 2 ) ≤ • • • ≤ n(e k ). Then (n(e 1 ), n(e 2 ), . . . , n(e k )) is called the signature of v. The graph Γ is said to be girthbiregular if it is bipartite, and all of its vertices belonging to the same bipartition have the same signature.