- Ringel's conjecture states that K
_{13}(above) has a cyclic decomposition into 13 copies of any tree with 6 edges. - The above conjecture was proven in 2020 by Montgomery, Pokrovskiy, and Sudakov.
- According to A000055, there are 11 trees with 6 edges.
- According to A033472, there are 164 graceful labelings on trees with 6 edges.

- Each row in this table represents an isomorphic class of trees.
- Each cell in this table represents a particular graceful labeling.
- In a given row, the right half is a reflection of the left half, so it is shown with a different background color.
- Let's say that a
**sundial**is a drawing of a graceful labeling whereby the value of the vertex label picks a vertex in the standard drawing of K_{13}. - Vertex 0 is the right-most vertex, and the labels increase in a counter-clockwise direction.
- Note that rotating a sundial 13 times will perfectly cover the K
_{13}. - An example of a graceful labeling and its corresponding sundial are shown below.
- As an experimental visual aid, the center nodes and leaf nodes are highlighted in a different color. This may or may not be useful.

Clicking one of the sundials in the table will reveal its corresponding graceful labeling. Alternatively, click the button below to swap the images in the table.