# Some Smarandache conclusions of coloring properties of complete uniform mixed hypergraphs deleted some C-hyperedges by ProQuest

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```									   Scientia Magna
Vol. 5 (2009), No. 3, 76-86

Some Smarandache conclusions of coloring
properties of complete uniform mixed
hypergraphs deleted some C-hyperedges
Guobiao Zheng
Qinghai Institute for Nationalities Journal editorial board, Qinghai Xining, 810007

Abstract The upper and lower chromatic number of uniform mixed hypergraphs and C-
hyperedge and D-hyperedge contact with the inevitable. In general, the increase in the C-
hyperedge will increase lower chromatic number χH , an increase D-hyperedge will decrease
upper chromatic number χH . In this paper the relationship between C-hyperedge with the
upper chromatic number and lower chromatic number and some conclusions with respect to
mixed hypergraph are given.
Keywords A complete uniform mixed hypergraph, chromatic, upper chromatic, Smarand-
ache conclusions.

§1. Lemma and the basic concepts
Deﬁnition 1.1.[1] Let X = {x1 , x2 , · · · , xn } be a ﬁnite set, C = {C1 , C2 , · · · , Cl }, D =
{D1 , D2 , · · · , Dm } are two subset clusters of X, all of which Ci ∈ C to meet |Ci | ≥ 2, and all
Dj ∈ D to meet |Dj | ≥ 2. Then H = (X, C, D) is called a mixed hypergraph from X, and each
Ci ∈ C is called the C−hyperedges, and each Dj ∈ D is called the D−hyperedges. In particular,
HD = (X, D) is called a D hypergraph, the HC = (X, C) for C− hypergraph.
Deﬁnition 1.2.[2] On 2 ≤ l, m ≤ n = |X|, let
X   X
K(n, l, m) = (X, C, D) = (X,       ,   )
l   m
where |C| = n and |D| = m , then K(n, l, m) is called the complete (l, m)-uniform mixed
l
n

hypergraph with n vertex.
It is clear that for a given n, l, m, in a sense of the isomorphic existence just has one
K(n, l, m).
Deﬁnition 1.3.[3,4] For mixed hypergraph H = (X, C, D), the largest i among all existence
of strict i-Coloring known as the upper chromatic number H, said that for χH .  ¯
Deﬁnition 1.4.[5] For mixed hypergraph H = (X, C, D), if a i Partition X = {X1 , X2 , · · · ,
Xi } of vertex sets X satisfy:
1) For each C-hyperedge at least two vertices is allocated in the same block;
2) For each D-hyperedge at least two vertices is allocated in diﬀerent blocks.
The partition is called as a feasible partition of H.
Obviously, any strict i coloring of H corresponds with a strict i feasible partition, and vice
versa. they are equivalent. Therefore, we write one feasible partition of H or a strict i-coloring
c as: c = X1 X2 · · · Xi and ri (H) = ri is the total number of all feasible i partition.
Some Smarandache conclusions of coloring properties of complete uniform mixed
Vol. 5                          hypergraphs deleted some C-hyperedges                                     77

Deﬁnition 1.5.[6] Let S be a subset of the vertices set X of mixed hypergraph H =
(X, C, D), if the set does not contain any of the C-Hyperedge (D-Hyperedge) as a subset, then
it is called C stable or C independent (D stable or D independent).
X
Lemma 1.1.[7,8] Let mixed hypergraph H = (X,                r   , D), where 2 ≤ r ≤ n = n(H), then
Arbitrary a coloring of H meet condition

χ(H) = r − 1.

Deﬁnition 1.6.[9] mixed hypergraph for H = (X, C, D), if there is a mapping c : Y →
{1, 2, · · · , λ} that between subset Y ∈ X and λ colors {1, 2, · · · , λ}, and it meet following
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