I am learning about Permutohedron's and I am having trouble determining how vertices are connected by edges. Wikipedia tells us that the edge-graph of a permutohedron is the Cayley graph of adjacent transpositions. There are two pictures on Wikipedia. The first one shows vertex $1423$ connected to vertex $1324.$ By analyzing the rest of the picture, I believe the vertices are connected if the number $1$ is swapped with number $2$, number $2$ is swapped with number $3$, or number $3$ is swapped with number $4$. (Also, the colouring of the first image on wikipedia is incorrect.)
However, there is another picture on wikipedia that shows vertex $1423$ connected to $1243.$ This could not be connected because number $2$ is swapped with number $4$ because $2$ and $4$ are not adjacent. However, upon further analysis it seems as though the vertices are connected if position $1$ is swapped with position $2,$ position $2$ with position $3,$ or position $3$ with position $4$.
Which picture, if any, is correct? What is the correct way to connect vertices of a permutohedron?
