^{1}

^{2}

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A k-L(2,1)-labeling for a graph G is a function such that whenever and whenever u and v are at distance two apart. The λ-number for G, denoted by λ( G), is the minimum k over all k-L(2,1)-labelings of G. In this paper, we show that for or 11, which confirms Conjecture 6.1 stated in [X. Li, V. Mak-Hau, S. Zhou, The L(2,1)-labelling problem for cubic Cayley graphs on dihedral groups, J. Comb. Optim. (2013) 25: 716-736] in the case when or 11. Moreover, we show that if 1) either (mod 6), m is odd, r = 3, or 2) (mod 3), m is even (mod 2), r = 0.

Let

The L(2,1)-labeling problem was studied very extensively in the literature and has attracted much attention. Griggs and Yeh [

Let

Next, for each pair

Li et al. [

Conjecture 1. [

Shao et al. [

Theorem 1. [

Therefore, Conjecture 1 is still open for odd

In this paper, we show that

From the definition of the brick product graph, it is clear that

Fact 1.

In [

Lemma 1. [

Proposition 1. Let

By Theorem 1, we have

We use the pattern

Now, we consider the case

Proposition 2. Let

Similar to Proposition 1, we only need to consider the case

Case 1:

We use the following pattern

Case 2:

We use the following pattern

From Propositions 1 and 2, we have

Theorem 2. Let

In [

Theorem 3. Let

Moreover,

1) 3 divides

2) 6 divides

Furthermore, if neither 1) nor 2) is satisfied, then

However, Theorem 3 consider the condition that

Theorem 4. Let

Let

Proposition 3. Let

Let

induces a 5-L(2,1)-labeling of

Proposition 4. Let

Let

Then Q induces a 5-L(2,1)-labeling of

Proposition 5. Let

Let

times. Then Q induces a 5-L(2,1)-labeling of

Proposition 6. Let

Let

By observing the results of Propositions 3 - 6, we propose the following con- jecture:

Conjecture 2. Let

This work was supported by Applied Basic Research (Key Project) of Sichuan Province under grant 2017JY0095, Key Project of Sichuan Provincial Department of Education under grant 17ZA0079 and Automotive Creative Design Pilot Area of Chengdu University and Longquanyi District under grant 2015-CX00- 00010-ZF.

Zhang, X.J., Yang, H. and Li, H. (2017) L(2,1)-Labeling of the Brick Product Graphs. Journal of Applied Mathematics and Physics, 5, 1529-1536. https://doi.org/10.4236/jamp.2017.58126