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Abstract
A graph
G is called
prime
if the vertices of
G can be
assigned distinct labels
1 , 2 , … , | V ( G ) |
such that the labels on any two adjacent vertices are relatively prime. By showing that for
every even
n
≤ 2 . 4 6 8
× 1 0 9
there exists
s
∈ [ 1 , n
− 1 ]
such that both
n
+
s
and
2 n
+
s
are prime, we prove the generalized Peterson graph
P ( n , 1 ) is prime for all even
n
∈ [ 4 , 2 . 4 6 8
× 1 0 9 ] . Moreover, for a
fixed
n we describe a
method for labeling
P ( n , k )
that is a prime labeling for multiple values of
k . Using this
method, we prove
P ( n , k ) is
prime for all even
n
≤ 5 0
and all odd
k
∈ [ 1 , n ∕ 2 ) .
Keywords
graph labeling, generalized Petersen graph, prime graph
Mathematical Subject Classification 2010
Primary: 05C78
Milestones
Received: 8 September 2015
Revised: 20 November 2015
Accepted: 28 November 2015
Published: 11 October 2016
Communicated by Joseph A. Gallian