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Large minimal irregular digraphs

Dziechcińska-Halamoda Z., Majcher Z., Michael J., Skupień Z.

Opuscula Mathematica

2003

Vol. 23

21-24

A digraph is called irregular if its distinct vertices have distinct degree pairs. An irregular digraph is called minimal if the removal of any arc results in a non-irregular digraph. A large minimal irregular digraph Fn of order n is constructed if n is the sum of initial positive integers. It is easily seen that the minimum and maximum sizes among n-vertex irregular digraphs are asymptotic to [formula] and n2, respectively. It appears that the size of Fn is asymptotic to n2, too. Similarly, a minimal irregular oriented graph Hn is constructed such that the size of Hn is asymptotic to 1/2n2 whence it is asymptotically the largest size among n-vertex oriented graphs whether irregular or not.