Labelling graphs with a condition at distance 2
SIAM Journal on Discrete Mathematics
Relating path coverings to vertex labellings with a condition at distance two
Discrete Mathematics
On the $\lambda$-Number of $Q_n$ and Related Graphs
SIAM Journal on Discrete Mathematics
On generalized Petersen graphs labeled with a condition at distance two
Discrete Mathematics
On Regular Graphs Optimally Labeled with a Condition at Distance Two
SIAM Journal on Discrete Mathematics
L(2, 1)-labeling of direct product of paths and cycles
Discrete Applied Mathematics - Structural decompositions, width parameters, and graph labelings (DAS 5)
Optimal L(d,1)-labelings of certain direct products of cycles and Cartesian products of cycles
Discrete Applied Mathematics
Real Number Graph Labellings with Distance Conditions
SIAM Journal on Discrete Mathematics
L(2, 1)-labelings of Cartesian products of two cycles
Discrete Applied Mathematics
The L(h, k)-Labelling Problem: A Survey and Annotated Bibliography
The Computer Journal
Exact λ-numbers of generalized Petersen graphs of certain higher-orders and on Möbius strips
Discrete Applied Mathematics
On L(2,1)-labeling of generalized Petersen graphs
Journal of Combinatorial Optimization
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In the classical channel assignment problem, transmitters that are sufficiently close together are assigned transmission frequencies that differ by prescribed amounts, with the goal of minimizing the span of frequencies required. This problem can be modeled through the use of an L(2,1)-labeling, which is a function f from the vertex set of a graph G to the non-negative integers such that |f(x)-f(y)|= 2 if xand y are adjacent vertices and |f(x)-f(y)|=1 if xand y are at distance two. The goal is to determine the @l-number of G, which is defined as the minimum span over all L(2,1)-labelings of G, or equivalently, the smallest number k such that G has an L(2,1)-labeling using integers from {0,1,...,k}. Recent work has focused on determining the @l-number of generalized Petersen graphs (GPGs) of order n. This paper provides exact values for the @l-numbers of GPGs of orders 5, 7, and 8, closing all remaining open cases for orders at most 8. It is also shown that there are no GPGs of order 4, 5, 8, or 11 with @l-number exactly equal to the known lower bound of 5, however, a construction is provided to obtain examples of GPGs with @l-number 5 for all other orders. This paper also provides an upper bound for the number of distinct isomorphism classes for GPGs of any given order. Finally, the exact values for the @l-number of n-stars, a subclass of the GPGs inspired by the classical Petersen graph, are also determined. These generalized stars have a useful representation on Moebius strips, which is fundamental in verifying our results.