Simply Sequentially Additive Labeling of Some Special Trees

Simply sequentially additive labeling 6509
f(w1ui) = 2i+9 ; 1 ≤ i ≤ n
n
f(w2vk,j) = 2n+6k+j+6 ; 1 ≤ k ≤ 3 , 1≤ j ≤ 3
Subcase 2.2: n ≡ 1 (mod 3)
Let V(G) = V1 ∪ V2 ∪ V3 ∪ V4 , where V1 = {wi ; 0 ≤ i ≤ n}, V2 = {ui ; 1 ≤ i ≤ n} ,
n
V3 = {vk,j ; 1 ≤ k ≤ [] 3
, 1 ≤ j ≤ 3}, V4 = {vk,1 ; k = 3
and E(G) = E1 ∪ E2 ∪ E3 ∪ E4 , where
E1 = {w0wi ; 1 ≤ i ≤ 4 } , E2 ={w1ui ; 1 ≤ i ≤ n } ,
E3 ={w2vk,j ; 1 ≤ k ≤ []
3
n+ 2 }
n , 1 ≤ j ≤ 3 } and E4 = {w2vk,1 ; k = 3
For 0 ≤ i ≤ 4, assign the label to wi and w0wi as in subcase 2.1.
⎧11
f(ui ) = ⎨
⎩2(i
+ 5)
;
;
i = 1
2 ≤ i ≤ n
⎧2n
+ 6k + j + 5 ;
f(vk,
j)
= ⎨
⎩10;
For 0 ≤ i ≤ 4
n-1
1 ≤ k ≤ 3 ; 1 ≤ j ≤ 3
n+
2 k = 3 ; j = 1
⎧12
;
f(w1 u i ) = ⎨
⎩ 2i + 11 ;
i = 1
2 ≤ i ≤ n
⎧2n
+ 6k + j + 8 ;
f(w 2v
k, j)
= ⎨
⎩13;
n-1
1 ≤ k ≤ 3
n+
2 k = 3
; 1 ≤ j ≤ 3
; j = 1
Subcase 2.3: n ≡ 2 (mod 3)
Let V(G) and E(G) be defined as in case 1.
f(w0) = n+3 ,f(w1) = 1, f(w2) = n + 5 , f(w3) = 4 , f(w4) = 5
⎧2
; i = 1
⎪
⎪
2(i + 1) ; 2 ≤ i ≤
+
f(u ) = ⎨2n
+ 9 ; i =
⎪
+
2n + 12; i = 2 ⎪
⎩(5n
+ 2i + 21) 2 ;
3 n
2 1 n
i
f(w0w1) = n + 4 , f(w0w2) = 2n+8,
f(w0w3) = n + 7 , f(w0w4) = n + 8 ,
f(w 2 i
⎧2n
+ 11
v ) = ⎨
⎩2n
+ i + 12
;
;
2 1 - n
n+
5
2
⎧3
; i = 1
⎪
⎪
2i + 3 ; 2 ≤ i ≤
f(w u ) = ⎨2n
+ 10 ; i =
⎪
+
2n + 13;
i =
⎪
⎩(5n
+ 2i + 23) 2 ;
n
n
1
i
+
2 1
2 1 - n
≤ i ≤ n
2
n+
5
2
3
≤ i ≤ n
i = 1
2 ≤ i ≤ n
)
f(v i
⎧n
+ 6
= ⎨
⎩n
+ i + 7
n+ 2 }
;
;
i = 1
2 ≤ i ≤ n