2015/07/19 by Cyriac Grigorious, Sudeep Stephen, Grigorious, Cyriac +7
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO
paper · pdf · doi:10.48550/arxiv.1507.05239
An error in the paper
arxiv created 2016/08/19 · arxiv updated 2016/10/30
For a vertex v of a connected graph G(V,E) and a subset S of V, the distance between v and S is defined by d(v,S)=min\d(v,x):x ∈ S \. For an ordered k-partition Π=\S1,S2… Sk\ of V, the representation of v with respect to Π is the k-vector r(v|Π) =(d(v,S1),d(v,S2)… d(v,Sk)). The k-partition Π is a resolving partition if the k-vectors r(v|Π), v ∈ V are distinct. The minimum k for which there is a resolving k-partition of V is the partition dimension of G. Salman et al.\rm\citeSaJaCh12 claimed that partition dimension of a class of circulant graphs C(n,± \1,2\), for all even n≥6 is 4 and it is 3 when n is odd. In this paper we obtain the partition dimension of circulant graphs G=C(n, ± \1,2 … j\), 1≤ j < \lfloor (n)/(2)\rfloor, n ≥(j+k)(j+1), n ≡ k mod (2j) and k and 2j are co-primes as, pd(G) = j+1 when j is even and all k=2m-1, 1 ≤ m ≤ j
pd(G)= j+1 when j is odd and all k=2m, 1 ≤ m ≤ j.