  We study critical bond percolation on periodic four-dimensional (4D) and
five-dimensional (5D) hypercubes by Monte Carlo simulations. By classifying the
occupied bonds into branches, junctions and non-bridges, we construct the
whole, the leaf-free and the bridge-free clusters using the
breadth-first-search algorithm. From the geometric properties of these
clusters, we determine a set of four critical exponents, including the thermal
exponent $y_{\rm t} \equiv 1/\nu$, the fractal dimension $d_{\rm f}$, the
backbone exponent $d_{\rm B}$ and the shortest-path exponent $d_{\rm min}$. We
also obtain an estimate of the excess cluster number $b$ which is a universal
quantity related to the finite-size scaling of the total number of clusters.
The results are $y_{\rm t} = 1.461(5)$, $d_{\rm f} = 3.044 \, 6(7)$, $d_{\rm B}
= 1.984\,4(11)$, $d_{\rm min} = 1.604 \, 2(5)$, $b = 0.62(1)$ for 4D; and
$y_{\rm t} = 1.743(10)$, $d_{\rm f} = 3.526\,0(14)$, $d_{\rm B} =
2.022\,6(27)$, $d_{\rm min} = 1.813\, 7(16)$, $b = 0.62(2)$ for 5D. The values
of the critical exponents are compatible with or improving over the existing
estimates, and those of the excess cluster number $b$ have not been reported
before. Together with the existing values in other spatial dimensions $d$, the
$d$-dependent behavior of the critical exponents is obtained, and a local
maximum of $d_{\rm B}$ is observed near $d \approx 5$. It is suggested that, as
expected, critical percolation clusters become more and more dendritic as $d$
increases.
