On $$(s,t)$$ ( s, t ) -relaxed $$L(2,1)$$ L ( 2, 1 ) -labelings of the triangular lattice

Wensong Lin () and Benqiu Dai
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Wensong Lin: Southeast University
Benqiu Dai: Southeast University

Journal of Combinatorial Optimization, 2015, vol. 29, issue 3, No 8, 655-669

Abstract: Abstract Suppose $$G$$ G is a graph. Let $$u$$ u be a vertex of $$G$$ G . A vertex $$v$$ v is called an $$i$$ i -neighbor of $$u$$ u if $$d_G(u,v)=i$$ d G ( u , v ) = i . A $$1$$ 1 -neighbor of $$u$$ u is simply called a neighbor of $$u$$ u . Let $$s$$ s and $$t$$ t be two nonnegative integers. Suppose $$f$$ f is an assignment of nonnegative integers to the vertices of $$G$$ G . If the following three conditions are satisfied, then $$f$$ f is called an $$(s,t)$$ ( s , t ) -relaxed $$L(2,1)$$ L ( 2 , 1 ) -labeling of $$G$$ G : (1) for any two adjacent vertices $$u$$ u and $$v$$ v of $$G,\,f(u)\not =f(v)$$ G , f ( u ) ≠ f ( v ) ; (2) for any vertex $$u$$ u of $$G$$ G , there are at most $$s$$ s neighbors of $$u$$ u receiving labels from $$\{f(u)-1,f(u)+1\}$$ { f ( u ) − 1 , f ( u ) + 1 } ; (3) for any vertex $$u$$ u of $$G$$ G , the number of $$2$$ 2 -neighbors of $$u$$ u assigned the label $$f(u)$$ f ( u ) is at most $$t$$ t . The minimum span of $$(s,t)$$ ( s , t ) -relaxed $$L(2,1)$$ L ( 2 , 1 ) -labelings of $$G$$ G is called the $$(s,t)$$ ( s , t ) -relaxed $$L(2,1)$$ L ( 2 , 1 ) -labeling number of $$G$$ G , denoted by $$\lambda ^{s,t}_{2,1}(G)$$ λ 2 , 1 s , t ( G ) . It is clear that $$\lambda ^{0,0}_{2,1}(G)$$ λ 2 , 1 0 , 0 ( G ) is the so called $$L(2,1)$$ L ( 2 , 1 ) -labeling number of $$G$$ G . In this paper, the $$(s,t)$$ ( s , t ) -relaxed $$L(2,1)$$ L ( 2 , 1 ) -labeling number of the triangular lattice is determined for each pair of two nonnegative integers $$s$$ s and $$t$$ t . And this provides a series of channel assignment schemes for the corresponding channel assignment problem on the triangular lattice.

Keywords: Channel assignment; $$L$$ L (2; 1)-labeling; ( $$s; t$$ s; t )-relaxed $$L$$ L (2; 1)-labeling; Triangular lattice (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10878-013-9615-y

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