In graph theory, the shift graph Gn,k for
n , k ∈
,
n > 2 k > 0
{\displaystyle n,k\in \mathbb {N} ,\ n>2k>0}
is the graph whose vertices correspond to the ordered
k
and where two vertices
are adjacent if and only if
Shift graphs are triangle-free, and for fixed
their chromatic number tend to infinity with
[1] It is natural to enhance the shift graph
with the orientation
be the resulting directed shift graph.
Note that
is the directed line graph of the transitive tournament corresponding to the identity permutation.
is the directed line graph of
The shift graph
is the line-graph of the complete graph
in the following way: Consider the numbers from
ordered on the line and draw line segments between every pair of numbers.
Every line segment corresponds to the
-tuple of its first and last number which are exactly the vertices of
Two such segments are connected if the starting point of one line segment is the end point of the other.