1. Introduction
Throughout this paper, we consider connected graphs without loops or multiple edges. For a graph 
and 
are used to denote the vertex set and edge set of 
and 
denote the minimum degree and the maximum degree of a graph G, respectively. For a vertex
, the neighborhood of v in G is 
is adjacent to v in
. Vertices in 
are called neighbors of v, 
denotes the number of vertices in
. The other terminology and notations are referred to [1].
For a given graph G, an integer
, an 
- labeling of G is defined as a function 
such that 
if
; and 
if
, where
, the distance of u and v, is the length (number of edges) of a shortest path between u and v. the 
-labeling number, denoted
, is the least integer 
such that G has a 
-labeling.
The Motivated by the channel assignment problem introduced by Hale in [2], the 
labeling have been studied extensively in the past decade. In 1992, in [3] Griggs and Yeh proposed the famous conjecture, for any graph
.
Griggs and Yeh in [3] proved that the conjecture true fop path, tree, circle, wheel and the graph with diameter 2, G. J. chang and David Kuo in [4] proved that 
for any graph. Recently Kral D and Skrekovski R in [5] proved the upper is
. It is difficult to prove the conjecture. Now, the study of 
- labeling is focus on special graph. Georges [6,7] give some good results. Zhang and Ma studied the labeling of some special graph, giving some good results in [8-11].
In this paper, we studied the 
-labeling number of the product and the join graph on two fans.
2. 
-Labeling Number of the Join Graph on Two Fans
Definition 2.1 Let 
be a fan with m + 1 vertices
, in which
.
Definition 2.2 Let G and H be two graphs, the join of G and H denoted
, is a graph obtained by starting with a disjoint union of G and H, and adding edges joining each vertex of G to each vertex of H.
Theorem 2.1 Let
, if
, then
.
Proof. In
, for arbitrary vertex u and v, such that
, clearly
.
Let k denote the maximum labeling number of 
First, we give a 
-labeling of 
as follows,
.
If 
when
,
when
,
when
,
when
.
If
, let
, 
,
.
If
, let
, 
,
.
If
, let
, 
,
.
If
, let
,
.
Clearly,
.
Then we label the vertex of 
as followsIf 
,
when
,
when
,
when
,
when
.
If
, let
If
, let
If
, let
If
, let
From aboveIf
,
is the maximum number in
, and
, then
If
,
is the maximum number in
, and
, then
If
,
is the maximum number in
, and
, then
If
,
is the maximum number in
, and
, then
So 
is the maximum number in
, and
, and
.
Obviously, f is a 
-
-labeling of GThen
.
3. 
-Labeling Number of the Product Graph on Two Fans
Definition 3.1 The Cartesian product of graph G and H, denoted
, which vertex set and edge set are the follows:
Theorem 3.1 Let
, if
, then
.
Proof. In
, the other vertices 
, In
, the other vertices
denote the vertex of
, Obviously, 
, for
.
We give a 
-labeling of G as follows, First, let
We have the maximum labeling number is 2n + 3.
Then let
From above, 
is the maximum labeling number.
Finally, let 
Obviously, 
is the maximum labeling number in these 
since n ≤ m < 2n, then the maximum labeling number no more than
, and
, so
.