The rectilinear local crossing number of a simple graph 
,
denoted 
,
is the minimum, over all straight
line drawings of 
in the plane, of the largest number
of other edges crossing the interior of a single graph edge. The vertices
are required to be in general position, meaning
that no three are collinear. The value is 0 when 
has no edges.
Ábrego and Fernández-Merchant (2017) determined the rectilinear local crossing number of the complete graph 
for every positive integer 
to be
![lcr^_(K_n)={4 for n=8; 15 for n=14; [1/2(n-3-[(n-3)/3])[(n-3)/3]] otherwise,](/images/equations/RectilinearLocalCrossingNumber/NumberedEquation1.svg) |
(1)
|
where ![[x]](/images/equations/RectilinearLocalCrossingNumber/Inline7.svg)
denotes the ceiling function. The sequence
given by the ceiling expression is 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 8, 9, 12, 14, ...
(OEIS A374243). It agrees with the rectilinear
local crossing numbers except at 
and 
, where the values are 4 and 15, respectively.
." J. Combin. Th. Ser. A 151, 131-145,
2017.