Ecclesiastical ring gold, stone-set ring. Roman Catholic bishops traditionally have their episcopal ring set with an amethyst. Aside from the rings a bishop purchases or is given by others, his rings belong to the Church; he will have inherited the previous bishop's ring collection, which is held in trust. While all hierarchs are accorded the honor of being buried wearing a ring, all rings belonging to the Church will be returned to the Church upon the retirement or death of any hierarch. In a Decree of Pope Boniface IV (A.D. 610) it describes monks raised to the episcopal dignity as "anulo pontificali
Uniquely colorable graph A uniquely total colorable graph is a "k"-total-chromatic graph that has only one possible (proper) "k"-total-coloring up to permutation of the colors. Empty graphs, paths, and cycles of length divisible by 3 are uniquely total colorable graphs. Some properties of a uniquely "k"-total-colorable graph "G" with "n" vertices: Here χ″("G") is the total chromatic number; Δ("G"), maximum degree; and δ("G"), minimum degree. Uniquely colorable graph In graph theory, a uniquely colorable graph is a k-chromatic graph that has only one possible (proper) "k"-coloring up to permutation of the colors. A complete graph is uniquely colorable, because the only proper coloring