Shift graph

In graph theory, the shift graph $G_{n,k}$ for $$ n,k \in \mathbb{N},\ n > 2k > 0 $$ is the graph whose vertices correspond to the ordered $$k$$-tuples $$ a = (a_1, a_2, \dotsc, a_k)$$ with $$1 \leq a_1 < a_2 < \cdots < a_k \leq n $$ and where two vertices $$ a, b $$ are adjacent if and only if $$a_i = b_{i+1}$$ or $$a_{i+1} = b_i$$ for all $$ 1 \leq i \leq k-1 $$. Shift graphs are triangle-free, and for fixed $$k$$ their chromatic number tend to infinity with $$n$$. It is natural to enhance the shift graph $$G_{n,k}$$ with the orientation $$a \to b$$ if $$a_{i+1}=b_i$$ for all $$1\leq i\leq k-1$$. Let $$\overrightarrow{G}_{n,k}$$ be the resulting directed shift graph. Note that $$\overrightarrow{G}_{n,2}$$ is the directed line graph of the transitive tournament corresponding to the identity permutation. Moreover, $$\overrightarrow{G}_{n,k+1}$$ is the directed line graph of $$\overrightarrow{G}_{n,k}$$ for all $$k \geq 2$$.

Further facts about shift graphs

 * Odd cycles of $$G_{n,k}$$ have length at least $$2k+1$$, in particular $$G_{n,2}$$ is triangle free.
 * For fixed $$k \geq 2$$ the asymptotic behaviour of the chromatic number of $$G_{n,k}$$ is given by $$\chi(G_{n,k}) = (1 + o(1))\log\log\cdots\log n $$ where the logarithm function is iterated $${\displaystyle k-1}$$ times.
 * Further connections to the chromatic theory of graphs and digraphs have been established in.
 * Shift graphs, in particular $$G_{n,3}$$ also play a central role in the context of order dimension of interval orders.

Representation of shift graphs
The shift graph $$G_{n,2}$$ is the line-graph of the complete graph $$K_n$$ in the following way: Consider the numbers from $$1$$ to $$n$$ ordered on the line and draw line segments between every pair of numbers. Every line segment corresponds to the $$2$$-tuple of its first and last number which are exactly the vertices of $$G_{n,2}$$. Two such segments are connected if the starting point of one line segment is the end point of the other.

Note: This seems false, since $$\{1,2\}$$ and $$\{1,3\}$$ will be non-adjacent. Someone should check this.