By Dileepkumar R
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Vt+1,n respectively and the centre of the web as v0,0 . Define a labeling f : V (W (t, n)) −→ N such that F (u) = f (u) if u ∈ V (G) 0 if u = v 30 f (vm,i ) = i−1 ( 2 )d + (m − 1)nd n−1 i ( 2 )d + ( 2 )d + (m − 1)nd ( 3n−1 )d + ( i−1 )d + (m − 2)nd 2 2 (n − 1)d + ( 2i )d + (m − 2)nd n−1 ( 2 )d + 2tnd m odd, i odd; 1 ≤ m ≤ t + 1, 1≤i≤n m odd, i even; 1 ≤ m ≤ t + 1, 2≤i≤n−1 m even, i odd; 2 ≤ m ≤ t + 1, 1≤i≤n m even, i even; 2 ≤ m ≤ t + 1, 2≤i≤n−1 (4) m = i = 0. Because f is injective on every cycle and the maximum vertex value in the mth cycle is less than the minimum vertex value in the (m + 1)th cycle, it is not hard to verify that f defined above is injective.
1 For all positive integer d and odd n, the generalized web graph W(t, n) is (k, d)-arithmetic for k = (n−1) d. v1,n . vt,n . vt+1,n respectively and the centre of the web as v0,0 . Define a labeling f : V (W (t, n)) −→ N such that F (u) = f (u) if u ∈ V (G) 0 if u = v 30 f (vm,i ) = i−1 ( 2 )d + (m − 1)nd n−1 i ( 2 )d + ( 2 )d + (m − 1)nd ( 3n−1 )d + ( i−1 )d + (m − 2)nd 2 2 (n − 1)d + ( 2i )d + (m − 2)nd n−1 ( 2 )d + 2tnd m odd, i odd; 1 ≤ m ≤ t + 1, 1≤i≤n m odd, i even; 1 ≤ m ≤ t + 1, 2≤i≤n−1 m even, i odd; 2 ≤ m ≤ t + 1, 1≤i≤n m even, i even; 2 ≤ m ≤ t + 1, 2≤i≤n−1 (4) m = i = 0.
1 The graph Ka,b,1 , a ≤ b, is (k+2r, k)-arithmetic for all integers k ≥ 1 and r ≥ 0. Proof Define a map f : A ∪ B ∪ C −→ N by f (w) = r f (vj ) = kj + r, 1≤j≤b f (ui ) = (b + 1)ki + r, 1 ≤ i ≤ a 43 can be easily verified to be a required (k+2r,k)-arithmetic numbering of Ka,b,1 . 5, where k = k + 2r and d = k for integers k ≥ 1 and r ≥ 0. 5 For the class of complete graphs in general we have the following conjecture to propose: Conjecture 2 For any integer n ≥ 5, Kn is not arithmetic. 2 there exists a (k, d)-arithmetic numbering f of C4t , t ≥ 1, such that k1 , k2 ∈ f (C4t ).
Arithmetic Graphs by Dileepkumar R
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