L(2,1)-labeling number of the product and the join graph on two fans are discussed in this paper, we proved that
L(2,1)-labeling number of the product graph on two fans is λ(G) ≤ Δ+3 ,
L(2,1)-labeling number of the join graph on two fans is λ(G) ≤ 2Δ+3.
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. <img src="15-7401498\888d1536-ee4e-43f6-b6aa-1985dc9f8e41.jpg" />-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. <img src="15-7401498\c106e952-bd68-4738-b15b-778198ea115a.jpg" />-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.
REFERENCESReferencesJ. A. Bondy and U. S. R. Murty, “Graph Theory with Applications,” Macmillan, New York, 1976.W. K. Hale, “Frequency Assignment: Theory and Appli cations,” IEEE Proceedings, Vol. 68, No. 12, 1980, pp. 1497-1514. doi:10.1109/PROC.1980.11899J. R. Griggs and R. K. Yeh, “Labeling Graphs with a Condition at Distance 2,” SIAM Journal on Discrete Ma thematics, Vol. 5, No. 4, 1992, pp. 586-595.
doi:10.1137/0405048G. J. Chang and D. Kuo, “The L(2,1)-Labeling Problem on Graphs,” SIAM Journal on Discrete Mathematics, Vol. 9, No. 2, 1996, pp. 309-316.
doi:10.1137/S0895480193245339D. Král and R. A. skrekovski, “Theorem about the Chan nel Asscgnment Problem,” SIAM Journal on Discrete Mathematics, Vol. 16, No. 3, 2003, pp. 426-437.
doi:10.1137/S0895480101399449J. P. Georges, D. W. Mauro and M. I. Stein, “Labeling Products of Complete Graphs with a Condition at Dis tance Two,” SIAM Journal on Discrete Mathematics, Vol. 14, No. 1, 2000, pp. 28-35.
doi:10.1137/S0895480199351859J. P. Georges, D. W. Mauro and M. A. Whittlesey, “Re lating Path Covering to Vertex Labelling with a Condi tion at Distance Two,” Discrete Mathematics, Vol. 135, 1994, pp. 103-111. doi:10.1016/0012-365X(93)E0098-OS. M. Zhang and Q. L. Ma, “On List (2,1)-Labelling of Some Planar Graphs,” Ars Combinatoria, Vol. 84, 2007, pp. 231-241.S. M. Zhang and Q. L. Ma, “Labelling Some Planar Graphs with a Condition at Distance Two,” Journal of Applied Mathematics and Computing, Vol. 24, No. 1-2, 2007, pp. 421-426.S. M. Zhang and J. H. Wang, “L(p,q)-Labeling of Planar Graph with High Maximum Degree,” Journal of Shan dong University, Vol. 42, No. 4, 2007, pp. 39-43.S. M. Zhang and Q. L. Ma, “L(d,1)-Total Labeling of Ou terplannar Graphs,” Journal of Jinnan University, Vol. 20, No. 3, 2006, pp. 258-260.
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 [
, 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.
labeling have been studied extensively in the past decade. In 1992, in [
.
for any graph. Recently Kral D and Skrekovski R in [
. 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].
-labeling number of the product and the join graph on two fans.
be a fan with m + 1 vertices
, in which
.
, 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.
, if
, then
.
, for arbitrary vertex u and v, such that
, clearly
.
-labeling of 
as follows,
.
when
,
when
,
when
,
when
.
, let
, 
,
.
, let
, 
,
.
, let
, 
,
.
, let
,
.
.
as followsIf 
,
when
,
when
,
when
,
when
.
, let
, let
, let
, let
,
is the maximum number in
, and
, then
,
is the maximum number in
, and
, then
,
is the maximum number in
, and
, then
,
is the maximum number in
, and
, then
is the maximum number in
, and
, and
.
-
-labeling of GThen
.
, which vertex set and edge set are the follows:
, if
, then
.
, the other vertices 
, In
, the other vertices
, Obviously, 
, for
.
-labeling of G as follows, First, let
is the maximum labeling number.
Obviously, 
is the maximum labeling number in these 
since n ≤ m < 2n, then the maximum labeling number no more than
, and
, so
.