This has been proved by . For certain graphs, even fewer than Δ colors may be needed. shows that Δ − 1 colors suffice if and only if the given graph has no Δ-clique, "provided" Δ is large enough. For triangle-free graphs, or more generally graphs in which the neighborhood of every vertex is sufficiently sparse, O(Δ/log Δ) colors suffice. The degree of a graph also appears in upper bounds for other types of coloring; for edge coloring, the result that the chromatic index is at most Δ + 1 is Vizing's theorem. An extension of Brooks' theorem to total coloring, stating that the total chromatic number is at most Δ + 2, has been conjectured by Mehdi Behzad and Vizing. The Hajnal–Szemerédi theorem on equitable coloring states that any graph has a (Δ + 1)-coloring in which the sizes of any two color classes differ by at most one. A Δ-coloring, or even a Δ-list-coloring, of a degree-Δ graph may be found in linear time. Efficient algorithms are also known for finding Brooks colorings in parallel and distributed models of computation.