Search results for "Oriented graph"
showing 3 items of 3 documents
The minimum size of fully irregular oriented graphs
2001
Abstract Digraphs in which any two vertices have different pairs of semi-degrees are called fully irregular. For n-vertex fully irregular oriented graphs (i.e. digraphs without loops or 2-dicycles) the minimum size is presented.
A smallest irregular oriented graph containing a given diregular one
2004
AbstractA digraph is called irregular if its vertices have mutually distinct ordered pairs of semi-degrees. Let D be any diregular oriented graph (without loops or 2-dicycles). A smallest irregular oriented graph F, F=F(D), is constructed such that F includes D as an induced subdigraph, the smallest digraph being one with smallest possible order and with smallest possible size. If the digraph D is arcless then V(D) is an independent set of F(D) comprising almost all vertices of F(D) as |V(D)|→∞. The number of irregular oriented graphs is proved to be superexponential in their order. We could not show that almost all oriented graphs are/are not irregular.
Extremal Irregular Digraphs
2018
A digraph is called irregular if its distinct vertices have distinct degree pairs. An irregular digraph is called minimal (maximal) if the removal of any arc (addition of any new arc) results in a non-irregular digraph. It is easily seen that the minimum sizes among irregular n-vertex whether digraphs or oriented graphs are the same and are asymptotic to (√2/3) n3/2; maximum sizes, however, are asymptotic to n2 and n2/2, respectively. Let s stand for the sum of initial positive integers, s = 1, 3, 6, . . . . An oriented graph Hs and a digraph Fs, both large (in terms of the size), minimal irregular, and on any such s vertices, s ≥ 21, are constructed in [Large minimal irregular digraphs, Op…