Odd mean labeling of the graphs $P_{a,b}
Odd mean labeling of the graphs $P_{a,b}
Odd mean labeling of the graphs $P_{a,b}
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148 R. VASUKI AND A. NAGARAJAN<br />
and<br />
{<br />
f(xi,j,1 ) + 2(k − 1)m if k is odd, 3 ≤ k < 2r, k+1 ≤ i ≤ r<br />
2<br />
f(x i,j,k ) =<br />
f(x i,j,2 ) + 2(k − 2)m if k is even, 4 ≤ k < 2r, k+2 ≤ i ≤ r.<br />
2<br />
It can be verified that, <strong>the</strong> label <strong>of</strong> <strong>the</strong> edges <strong>of</strong> <strong>the</strong> graph are 1, 3, 5, . . . ,<br />
2r(r + 1)m − 1. Hence P 2t+1<br />
〈2r〉<br />
is an odd <strong>mean</strong> graph for all values <strong>of</strong> r and t.<br />
Case(ii) when m is even.<br />
Let m = 2t, t ∈ Z + .<br />
Define f on V (P 2t<br />
2r) as follows:<br />
f(y 1 ) = 0,<br />
f(y i ) = f(y i−1 ) + 2(2i − 2)m,<br />
f(y r+1 ) = 2r(r + 1)m − 1,<br />
f(x 1,j,1 ) = 4j − 2, 1 ≤ j ≤ m,<br />
f(x i,j,1 ) = f(y i ) + f(x 1,j,1 ), 2 ≤ i ≤ r, 1 ≤ j ≤ m,<br />
{<br />
f(yi ) + 4m + 4(i − 1), 1 ≤ i ≤ t<br />
f(x i,j,2 ) =<br />
f(y i ) + (2m + 3) + 4(i − (t + 1)),<br />
t + 1 ≤ i ≤ 2t<br />
and<br />
{<br />
f(xi,j,1 ) + 2(k − 1)m if k is odd, 3 ≤ k < 2r, k+1 ≤ i ≤ r<br />
2<br />
f(x i,j,k ) =<br />
f(x i,j,2 ) + 2(k − 2)m if k is even, 4 ≤ k < 2r, k+2 ≤ i ≤ r.<br />
2<br />
It is easy to check that, <strong>the</strong> label <strong>of</strong> <strong>the</strong> edges <strong>of</strong> <strong>the</strong> graph are 1, 3, 5, . . . ,<br />
2r(r + 1)m − 1. Hence P〈2r〉 2t is an odd <strong>mean</strong> graph for all values <strong>of</strong> r and t. Thus<br />
P〈2r〉 m is an odd <strong>mean</strong> graph for all values <strong>of</strong> r and m.<br />
For example, odd <strong>mean</strong> <strong>labeling</strong>s <strong>of</strong> <strong>the</strong> <strong>graphs</strong> P〈8〉 6 and P 〈8〉 5 are shown in Figure<br />
8. □