Theorem 4.11. Assuming that:

$d = 2 l$ large enough

Then there is

𝜀 = 𝜀_{d} > 0

such that there are

d

-regular graphs (or multi-graphs)

G_{n}

on

n

vertices with

Φ (G_{n}) \geq 𝜀

for all sufficiently large

n

.