Shift graph

In graph theory, the shift graph for is the graph whose vertices correspond to the ordered -tuples with and where two vertices are adjacent if and only if or for all. Shift graphs are triangle-free, and for fixed their chromatic number tend to infinity with. It is natural to enhance the shift graph with the orientation if for all. Let be the resulting directed shift graph.
Note that is the directed line graph of the transitive tournament corresponding to the identity permutation. Moreover, is the directed line graph of for all.

Further facts about shift graphs

Odd cycles of have length at least, in particular is triangle free.
For fixed the asymptotic behaviour of the chromatic number of is given by where the logarithm function is iterated times.
Further connections to the chromatic theory of graphs and digraphs have been established in.
Shift graphs, in particular also play a central role in the context of order dimension of interval orders.

Representation of shift graphs

The shift graph is the line-graph of the complete graph in the following way: Consider the numbers from to 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.