Indeed, for each color [c.sub.i], the 0-1 matrix of size n x m obtained by replacing [c.sub.i] by 1 and all other colors by 0 is an adjacency matrix of a simple bipartite graph such that the

disjoint union of these simple graphs is [K.sub.n,m].

Involutions fixing the

disjoint union of copies of H[P.sub.n], J.

For this purpose, we introduce a natural topology on Milnor's K-groups [K.sup.M.sub.l](k) for a topological field k as the quotient topology induced by the joint determinant map and show that, in case of k = R or C, the natural topology on [K.sup.M.sub.l](k) is

disjoint union of two indiscrete components or indiscrete topology, respectively.

A subset of R is [phi]-open if and only if it is a

disjoint union of maximal intervals (i.e., such that the union of any two or more of these intervals is not an interval itself) such that any of these intervals whose right endpoint is an integer is right-open.

The authors showed that G[u.sub., n] was the

disjoint union of isomorphic copies of their special subgraph [F.sub.u, n], the generalized Farey graph, coming from the use of the subgroup [[GAMMA].sub.0] (n) of [GAMMA].

Define an EDT0L system [H.sub.#] = ([V.sub.#],A [union] {#}, [[DELTA].sub.#],[I.sub.#]) as follows: [V.sub.#] is the

disjoint union V [union] {[c.sub.#] | c [member of] V}, [I.sub.#] = {[s.sub.#] | s [member of] I}, and m = ma[x.sub.P[member of][DELTA]]{|w| | (c, w) [member of] P}, the length of the longest right-hand side of any table.

The group G can be expressed by G = [G.sub.njp] [union] [G.sub.fp], where [union] denotes the

disjoint union.

For different values of d , the results related to a super ) , ( d a -EAT labeling can be found for w-trees [8], extended w-trees [9, 10], generalized extended w-trees [11, 12] , stars [20], subdivided stars [13, 14, 15, 21, 22, 29], path-like trees [2], caterpillars [17, 18, 25], subdivided caterpillar [16],

disjoint union of stars and books [6] and wheels, fans and friendship graphs [24], paths and cycles [23] and complete bipartite graphs [1].

As H is the

disjoint union of graphs [H.sub.1],..., [H.sub.m], an inductive application of Lemma 3.3 (selecting [S.sub.1] and [S.sub.2] to be empty) yields

The conjecture holds that any closed, orientable, prime 3-manifold M contains a

disjoint union of embedded incompressible 2-tori and Klein bottles such that each connected component of the complement admits a complete, locally homogeneous Riemannian metric of finite volume.

Then, the

disjoint union of arbitrary number of copies of G, that is, sG, s [greater than or equal to] 1, also admits a super (A',1)-EAT labeling.

The

disjoint union of [G.sub.1],[G.sub.2] is the graph [G.sub.1] [union] [G.sub.2] having as a vertex set the

disjoint union of V([G.sub.1]),V([G.sub.2]), and as an edge set the

disjoint union of E([G.sub.1]),E([G.sub.2]).