Zhendong Shao

The L(2,1)-labeling on planar graphs

Abstract Given non-negative integers j and k , an L ( j , k ) - labeling of a graph G is a function f from the vertex set V ( G ) to the set of all non-negative integers such that | f ( x ) − f ( y ) | ≥ j if d ( x , y ) = 1 and | f ( x ) − f ( y ) | ≥ k if d ( x , y ) = 2 . The L ( j , k ) -labeling number λ j , k is the smallest number m such that there is an L ( j , k ) -labeling with the largest value m and the smallest label 0. This paper presents upper bounds on λ 2 , 1 and λ 2 , 1 of a graph G in terms of the maximum degree of G for several classes of planar graphs. These bounds are the same as or better than previous results for the maximum degree less than or equal to 4.

Improved Bounds on the

The frequency assignment problem is to assign a frequency which is a nonnegative integer to each radio transmitter so that interfering transmitters are assigned frequencies whose separation is not in a set of disallowed separations. This frequency assignment problem can be modelled with vertex labelings of graphs. An <i>L</i>(2,1)-labeling of a graph <i>G</i> is a function <i>f</i> from the vertex set <i>V</i>(<i>G</i>) to the set of all nonnegative integers such that |<i>f</i>(<i>x</i>)-<i>f</i>(<i>y</i>)| ges 2 if <i>d</i>(<i>x</i>,<i>y</i>)=1 and |<i>f</i>(<i>x</i>)-<i>f</i>(<i>y</i>)| ges 1 if <i>d</i>(<i>x</i>,<i>y</i>)=2 , where <i>d</i>(<i>x</i>,<i>y</i>) denotes the distance between <i>x</i> and <i>y</i> in <i>G</i>. The <i>L</i>(2,1) -labeling number lambda(<i>G</i>) of <i>G</i> is the smallest number <i>k</i> such that <i>G</i> has an <i>L</i>(2,1)-labeling with max{<i>f</i>(<i>v</i>):<i>v</i> isin <i>V</i>(<i>G</i>)}=<i>k</i>. This paper considers the graph formed by the direct product and the strong product of two graphs and gets better bounds than those of Klavzar and Spacapan with refined approaches.

L(2,1)

An L(2,1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that | f (x) f (y)| 2 if d(x, y) = 1 and| f (x) f (y)| 1 if d(x, y) = 2, where d(x, y) denotes the distance between x and y in G. The L(2,1)-labeling number ( G) of G is the smallest number k such that G has an L(2,1)-labeling with max{ f (v): v2 V(G)} = k. Griggs and Yeh conjecture that ( G) 2 for any simple graph with maximum degree 2. In this work, we consider the total graph and derive its upper bound of ( G). The total graph plays an important role in other graph coloring problems. Griggs and Yeh’s conjecture is true for the total graph in some cases. c 2007 Elsevier Ltd. All rights reserved.

-Number of Direct and Strong Products of Graphs

The frequency assignment problem is to assign a frequency which is a nonnegative integer to each radio transmitter so that interfering transmitters are assigned frequencies whose separation is not in a set of disallowed separations. This frequency assignment problem can be modelled with vertex labelings of graphs. An L(2, 1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that |f(x)-f(y)|ges2 if d(x, y)=1 and |f(x)-f(y)|ges1 if d(x, y)=2, where d(x, y) denotes the distance between x and y in G. The L(2, 1)-labeling number lambda(G) of G is the smallest number k such that G has an L(2, 1)-labeling with max{f(v):visinV(G)}=k. In this paper, we develop a dramatically new approach on the analysis of the adjacency matrices of the graphs to estimate the upper bounds of lambda-numbers of the four standard graph products. By the new approach, we can achieve more accurate results and with significant improvement of the previous bounds.

The L (2, 1)-labeling on graphs and the frequency assignment problem

Abstract An L ( 2 , 1 ) -labeling of a graph G is a function f from the vertex set V ( G ) to the set of all nonnegative integers such that | f ( x ) − f ( y ) | ≥ 2 if d ( x , y ) = 1 and | f ( x ) − f ( y ) | ≥ 1 if d ( x , y ) = 2 , where d ( x , y ) denotes the distance between x and y in G . The L ( 2 , 1 ) -labeling number λ ( G ) of G is the smallest number k such that G has an L ( 2 , 1 ) -labeling with max { f ( v ) : v ∈ V ( G ) } = k . Griggs and Yeh conjecture that λ ( G ) ≤ Δ 2 for any simple graph with maximum degree Δ ≥ 2 . This paper considers the graph formed by the Cartesian sum of two graphs. As corollaries, the new graph satisfies the above conjecture (with minor exceptions).

A New Approach to the

Abstract An L ( 2 , 1 ) -labeling of a graph G is a function f from the vertex set V ( G ) into the set of nonnegative integers such that | f ( x ) − f ( y ) | ≥ 2 if d ( x , y ) = 1 and | f ( x ) − f ( y ) | ≥ 1 if d ( x , y ) = 2 , where d ( x , y ) denotes the distance between x and y in G . The L ( 2 , 1 ) -labeling number, λ ( G ) , of G is the minimum k where G has an L ( 2 , 1 ) -labeling f with k being the absolute difference between the largest and smallest image points of f . In this work, we will study the L ( 2 , 1 ) -labeling on K 1 , n -free graphs where n ≥ 3 and apply the result to unit sphere graphs which are of particular interest in the channel assignment problem.

L(2,1)

An L(2,1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that |f(x)-f(y)|>=2 if d(x,y)=1 and |f(x)-f(y)|>=1 if d(x,y)=2, where d(x,y) denotes the distance between x and y in G. The L(2,1)-labeling number @l(G) of G is the smallest number k such that G has an L(2,1)-labeling with max{f(v):v@?V(G)}=k. Griggs and Yeh conjecture that @l(G)@?@D^2 for any simple graph with maximum degree @D>=2. This paper considers the graph formed by the skew product and the converse skew product of two graphs with a new approach on the analysis of adjacency matrices of the graphs as in [W.C. Shiu, Z. Shao, K.K. Poon, D. Zhang, A new approach to the L(2,1)-labeling of some products of graphs, IEEE Trans. Circuits Syst. II: Express Briefs (to appear)] and improves the previous upper bounds significantly.

-Labeling of Some Products of Graphs

Abstract An L ( 2 , 1 ) -labeling of a graph G is a function f from the vertex set V ( G ) to the set of all nonnegative integers such that | f ( x ) − f ( y ) | ≥ 2 if d ( x , y ) = 1 and | f ( x ) − f ( y ) | ≥ 1 if d ( x , y ) = 2 , where d ( x , y ) denotes the distance between x and y in G . The L ( 2 , 1 ) -labeling number λ ( G ) of G is the smallest number k such that G has an L ( 2 , 1 ) -labeling with max { f ( v ) : v ∈ V ( G ) } = k . Griggs and Yeh conjecture that λ ( G ) ≤ Δ 2 for any simple graph with maximum degree Δ ≥ 2 . This work considers the graph formed by the skew product and the converse skew product of two graphs. As corollaries, the new graph satisfies the above conjecture (with minor exceptions).