CS 453 Graph Theory Assignment 6 solution

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1. Show that for any π‘˜ β‰₯ 3, if a tree 𝑇 has fewer than π‘˜ leaves, then the
maximum degree Ξ”(𝑇) among the vertices of 𝑇 must satisfy Ξ”(𝑇) < π‘˜. It
can help to consider the summations
𝑛 = βˆ‘π‘›π‘—
∞
𝑗=1
; 2(𝑛 βˆ’ 1) = total degree = βˆ‘π‘—π‘›π‘—
∞
𝑗=1
.
The phrase β€œπ‘‡ has fewer than π‘˜ leaves” means 𝑛1 < π‘˜.
The two sums can be combined into the single sum
βˆ‘(2 βˆ’ 𝑗)𝑛𝑗 = 2
𝑛
𝑗=1
It suffices to show that 𝑛𝑗 = 0 for all 𝑗 β‰₯ π‘˜.
2. Let (𝑇, π‘Ÿ) be a rooted tree. Recall that the level of a vertex π‘₯ is 𝐿(π‘₯) =
𝐷(π‘Ÿ, π‘₯). Also, the height of a rooted tree 𝐻 is the maximum of the levels of
its vertices.
a. Show that if π‘Ÿ is on the unique 𝑒, 𝑣-path, then 𝐷(𝑒, 𝑣) = 𝐿(𝑒) +
𝐿(𝑣).
b. Show that if 𝐿(𝑒) + 𝐿(𝑣) = 𝐷(𝑒, 𝑣), then π‘Ÿ must be on the unique
𝑒, 𝑣-path.
c. Show that for any two vertices 𝑒 and 𝑣, 𝐷(𝑒, 𝑣) ≀ 2𝐻.
d. Show that if 𝐷(𝑒, 𝑣) = 2𝐻, then 𝑒 and 𝑣 must be non-parents.
Equivalently, you can show that if either 𝑒 or 𝑣 is a parent, then
𝐷(𝑒, 𝑣) < 2𝐻.
3. Suppose (𝑇, π‘Ÿ) is a rooted π‘ž-ary tree where every parent has exactly π‘ž
children; such a tree is said to be saturated.
a. Show that 𝑇 has π‘π‘ž edges for some integer 𝑏.
b. Find a formula for the number of vertices of 𝑇 in terms of 𝑏, π‘ž.
c. Find a formula for the number of non-parents in terms of 𝑏, π‘ž.
4. Suppose (𝑇, π‘Ÿ) is a rooted tree with exactly 1012 edges. Recall that a lower
bound or an upper bound on 𝐻 is tight if there exists an example 𝑇 where
that bound is attained.
a. Find tight lower and upper bounds for 𝐻, the height of 𝑇.
b. Find tight lower and upper bounds for 𝐻 if 𝑇 is a saturated rooted
binary tree. Recall that saturated means every parent has the
maximum allowed number of children; here, that number is 2.