| 摘要: | 摘要： A variation of the classical channel assignment problem is to assign a radio channel which is a nonnegative integer to each radio transmitter so that “close” transmitters must receive different channels and “very close” transmitters must receive channels that are at least two channels apart. The goal is to minimize the span of a feasible assignment. This channel assignment problem can be modeled with distance-dependent graph labelings. A k - L (2,1)-labeling of a graph G is a mapping f from the vertex set of G to the set {0,1,2,…, k } such that | f ( x )− f ( y )|≥2 if d ( x , y )=1 and if d ( x , y )=2, where d ( x , y ) is the distance between vertices x and y in G . The minimum k for which G admits an k - L (2,1)-labeling, denoted by λ ( G ), is called the λ -number of G . Very little is known about λ -numbers of 3-regular graphs. In this paper we focus on an important subclass of 3-regular graphs called generalized Petersen graphs. For an integer n ≥3, a graph G is called a generalized Petersen graph of order n if and only if G is a 3-regular graph consisting of two disjoint cycles (called inner and outer cycles) of length n , where each vertex of the outer (resp. inner) cycle is adjacent to exactly one vertex of the inner (resp. outer) cycle. In 2002, Georges and Mauro conjectured that λ ( G )≤7 for all generalized Petersen graphs G of order n ≥7. Later, Adams, Cass and Troxell proved that Georges and Mauro’s conjecture is true for orders 7 and 8. In this paper it is shown that Georges and Mauro’s conjecture is true for generalized Petersen graphs of orders 9, 10, 11 and 12.
其他題名： J Comb Optim
出版者： Boston: Springer US
出版日期： 2012-10-01
出處： Journal of combinatorial optimization, 2012-10, Vol.24 (3), p.266-279
版權： Springer Science+Business Media, LLC 2011
版權： 2014 INIST-CNRS
識別號： ISSN: 1382-6905
識別號： EISSN: 1573-2886
識別號： DOI: 10.1007/s10878-011-9380-8 |
