Balance in Random Trees

We prove that a random labeled (unlabeled) tree is balanced. We also prove that random labeled and unlabeled trees are strongly $k$-balanced for any $k\geq 3$.

The case of k = 2 is especially interesting. In this case, the sets V 1 (c), V 2 (c), E 1 (c), E 2 (c) are called the sets of black vertices, white vertices, black edges, and white edges respectively. If the coloring c is fixed we may drop it in the notation. It is not difficult to see that: a) The complete graph K n is balanced iff n ≤ 3 or n is even.
b) The star S n is balanced iff n ≤ 5; see Fig.2 for a balanced coloring of S 5 .
c) The double star S p,q is balanced iff |p − q| ≤ 3. Figure 1. with the given coloring, the graph has 4 black and 3 white vertices; it also has 2 white edges (labeled with a "W") and 1 black edge (labeled with a "B") There are several other ways of defining balanced graphs which are similar to the one above but the definition we are using is the most interesting and challenging from the point of view of our problem.
In [Cah1], the author introduces a somewhat similar notion of a cordial graph, a generalization of both graceful and harmonious graphs. It has been conjectured by A.Rosa, G.Ringel and A.Kotzig that every tree is graceful (Graceful Tree Conjecture, [Ga]), and it has been conjectured by R. Graham and N.Sloane that every tree is harmonious (see [GS]). While these conjectures are still open, in [Cah2] it is proved that every tree is cordial.
Not every tree is balanced; in this paper, we will be interested in the property of being balanced for a random labeled and unlabeled tree, as well as for random labeled graphs.
The main results of the paper are Theorem A and Theorem B stated below.
Theorem A. A random labeled (unlabeled) tree is balanced; more precisely, if t n (τ n ) denotes the number of all labeled (unlabeled) trees on n vertices, and b n (b n ) denotes the number of all balanced labeled (unlabeled) trees on n vertices, then lim n→∞ b n t n = 1 and lim n→∞ b n τ n = 1.
Remark 1.3. In this paper, for simplicity, we consider only uniform models of random graphs and random trees. The results can be extended to a large class of non-uniform models as well. Note that t n = n n−2 (see [C] or [W]) and τ n ∼ Cα n n −5/2 for some positive constants C and α (see [O]).
We also would like to introduce the notion of k-balanced graphs.
The map c will be called a k-balanced coloring.
The map c will be called a strongly k-balanced coloring.
In more popular terms, a finite simple graph is strongly k-balanced iff it is k-equitably colorable. In Section 5 we study some basic properties of k-balanced graphs. We prove the following theorem.
Notes: 1. For any finite simple graph G, we will denote the maximal degree of G by d max (G).
2. A vertex of degree one will be called a leaf vertex or simply a leaf. A non-leaf vertex v is called a pre-leaf vertex if it is adjacent exactly to m − 1 leaves where m = deg(v). A pre-leaf vertex of degree two is called special.
3. For n ≥ 2, there exists a unique tree up to isomorphism with n vertices and maximal degree at most two; we will call this tree a string on n vertices.
4. For a tree G = (V, E) and a non-leaf vertex v ∈ V , a subset A ⊆ V will be called a branch of G with respect to v if there exists a vertex u adjacent to v such that A = {x ∈ V | d(x, u) < d(x, v)} where d(., .) denotes the distance in the tree G.

Characterization of Balanced Graphs
In this section we observe some basic facts on balanced and kbalanced graphs. Let us first prove a very simple lemma which provides a necessary and sufficient condition for a graph to be balanced.
Assume G is balanced with a balanced coloring c : V → {1, 2}.
Since G is balanced, we get |Card(I)−Card(J)| ≤ 1 so condition (i) is satisfied.
For every i ∈ I, we denote and for every j ∈ J, we denote

Thus condition (ii) is also satisfied.
To prove the converse, assume conditions (i) and (ii) are satisfied. We define the coloring c : V → {1, 2} as follows: for every i ∈ I we set c(v i ) = 0 and for every j ∈ J we set c(v j ) = 1.
Then we have Card(E 1 ) = 1 2 i∈I On the other hand, Then by condition (ii), we get |Card( Corollary 2.2. It is proved in [LLT] that an r-regular finite simple graph with n vertices is balanced iff n is even or r = 2. This fact also follows immediately from Lemma 2.1. In [KLST], the authors deduce the same fact from their characterization of balanced graphs. Lemma 2.1 shows that the balancedness of a graph totally depends on the degree sequence of it. This is no longer the case for k-balanced graphs for k ≥ 3. In fact, the trees G 1 and G 2 in Figure 3 have the same degree sequence (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 11), and it is not difficult to see that G 1 is 3-balanced while G 2 is not. Figure 3. The trees G 1 and G 2 have the same degree sequence; G 1 is 3-balanced while G 2 is not.
The fact that, for k ≥ 3, the k-balancedness is not determined by the degree sequence causes difficulties in proving that random graphs are k-balanced. It also seems plausible that, generically, k-balancedness is a weaker condition than balancedness, although it does not seem easy to describe (with a good sufficient condition) when exactly is this true. It is useful to point out the following simple fact.
Proposition 2.3. For all distinct m, n ≥ 2 there exists a finite simple graph which is m-balanced but not n-balanced.
Proof. Let p be a prime number such that p > max{m, n}.
Let us first assume that m > n. If n divides m, then the graph K m+1 is m-balanced but not n-balanced. If n does not divide m then the graph K mp is m-balanced but not n-balanced. Now assume that m < n. Then the graph K mp is m-balanced but not n-balanced.

Combinatorial Lemmas
The elements of M n consist of sequences of positive integers of length n such that no term is bigger than n. We denote max(d) = max 1≤i≤n d i .

Now we introduce the notion of balanced sequences
The partition {1, . . . , n} = I J will be called a balanced partition.
In these new terms, Lemma 2.1 states that a graph is balanced if and only if its degree sequence is balanced.
When the sequence is not balanced, we would like to measure how far it is from being balanced.
Definition 3.1. Letā = (a 1 , . . . , a n ) be any finite sequence of nonnegative integers. The quantity will be called the balance ofā.
Proof. We will present a constructive proof.
We divide the sequence into pairs (d 1 , d 2 ), . . . , (d 2m−1 , d 2m ), and we will abide by the rule that exactly one element of each pair belongs to I and the other element belongs to J. We start by letting If n is odd, then we may replaceā byā = (0, a 1 , . . . , a n ) and apply the previous argument.

Proof of Theorem A
First, we will discuss the case of labeled trees. The following theorem of J.W.Moon will play a crucial role Theorem 4.1 (See (M)). If > 0 is a fixed positive constant, then in a random labeled tree with n vertices, the maximal degree d max satisfies the following inequality (1 − ) log n log log n < d max < (1 + ) log n log log n Remark 4.2. By choosing = 0.1 we obtain that 0.9 log n log log n < d max < 1.1 log n log log n in a random tree with n vertices.
We will use only the upper bound in the inequality of Remark 4.2. Besides the upper bound on the maximal degree in random trees, we also need a lower bound on the number of vertices with degree 1, and with degree 2. Notice that, since the sum of degrees of a tree with n vertices is exactly 2n−2, at least half of the vertices have degree either 1 or 2. However, we need a linear lower bound for the number of vertices of degree 1 and for the number of vertices of degree 2 separately.
Let X i (T ), 1 ≤ i ≤ 2 be the random variable which denotes the number of vertices of degree i in a labeled tree T with n vertices. Also let µ = n e , σ 2 1 = n e (1 − 2 e ), σ 2 2 = n e (1 − 1 e ). It has been proved by A.Rényi (see [R]) that the asymptotic distribution of random variable is normal with mean µ and variance σ 2 1 . A similar result has been proved for the random variable X 2 −µ σ 2 , by A.Meir and J.W.Moon (see [MM]), namely, that the asymptotic distribution of the random variable X 2 −µ σ 2 is normal with mean µ and variance σ 2 2 . Combining these two results we can state the following theorem (due to A.Rényi and A.Meir-J.W.Moon) Theorem 4.3. Let α, β be fixed real numbers, α < β; and for i ∈ {1, 2}, let P i (α, β) denotes the probability that α < X i −µ σ 1 < β. Then We need the following immediate corollary of this theorem Corollary 4.4. In a random labeled tree with n vertices, for all i ∈ {1, 2}, X i ≥ 2 log n log log n . Now, in the case of random labeled trees, Theorem A immediately follows from Theorem 4.1, Corollary 4.4, and Lemma 3.5.
The case of unlabeled trees: We will use the results analogous to Theorem 4.1 and Theorem 4.3. The analogue of Theorem 4.1 is proved by W.Goh and E.Schmutz: Theorem 4.5 (See (GS)). There exists positive constants c 1 , c 2 such that in a random unlabeled tree with n vertices the maximum degree d max satisfies the inequality c 1 log(n) < d max < c 2 log(n). Now, for any k ∈ N let the random variable Y k denotes the number of vertices of degree k in a random unlabeled tree with n vertices. The following theorem is due to M.Drmota and B.Gittenberger; in the case of k ∈ {1, 2}, as a special case, it provides an analogue of Theorem 4.3.
Theorem 4.6 (See (DG)). For arbitrary fixed natural k, there exists positive constants µ k and σ k such that the limiting distribution of Y k is normal with mean µ(n) ∼ µ k n and variance σ(n) ∼ σ 2 k n.
Corollary 4.7. For all c > 0 and i ∈ 1, 2, in a random unlabeled tree with n vertices Y i > clog(n).
Now, in the case of unlabeled trees, the claim of Theorem A follows from Theorem 4.5, Lemma 3.5, and Corollary 4.7.

k-balanced trees: proof of Theorem B
In this section we will assume that k ≥ 3. The fact that the kbalancedness is not determined by the degree sequence causes significant difficulties in proving that random graphs are balanced. We nevertheless prove that random trees are strongly k-balanced by more careful study of k-balancedness.
First, we need to prove the following technical lemma.
Lemma 5.1. Let G = (V, E) be a tree and u, v be distinct vertices of G with degrees at least |G| 3 . Let also p, q be distinct pre-leaf vertices of G. Then there exists a strongly 3-balanced coloring c : V → {1, 2, 3} of G such that c(u) = c(v) and c(p) = c(q).
Proof. The proof is by induction on n = |G|. For n ≤ 5 the claim is obvious so we will assume that n ≥ 6 and the claim holds for all trees of order less than n.
Assume that at least one of the following two conditions hold: (c1) there exists z ∈ {p, q}\{u, v} such that deg(z) ≥ 3; (c2) there exists a leaf vertex not adjacent to any of the vertices u, v, p, q.
Then there exists a leaf w such that if G is a complete subgraph on V \{w}, then, in the tree G , we have min{deg(u), deg(v)} ≥ |G | 3 , and p, q are still pre-leaf vertices.
The following proposition is interesting in itself; it will also play a key role in proving Theorem B.
Proposition 5.2. Let G = (V, E) be a tree with n vertices where d max (G) ≤ n 3 . Then G is strongly 3-balanced. Moreover, for any two distinct pre-leaf vertices p and q of G there exists a strongly 3-balanced coloring c : V → {1, 2, 3} such that c(p) = c(q).
Proof. The proof will be by induction on n. For n ≥ 8 we have d max (G) ≤ 2 hence G is isomorphic to a string, thus the claim is obvious. Let us now assume that n ≥ 9, and the claim holds for all trees G of order less than n with d max (G ) ≤ |G | 3 . Let G = (V, E) and n = 3k + r, r ∈ {0, 1, 2}. We will consider the following three cases separately: Let v be a leaf of G, V = V \{v}, and let G = (V , E ) be the complete subgraph of G on V . Then we have By inductive hypothesis, there exists a strongly 3-balanced coloring c : V → {1, 2, 3} of G .
On the other hand, v is adjacent to exactly one vertex in G; let u be this vertex. Let j be any element of {1, 2, 3}\{c (u)}. We extend the coloring c of G to a strongly 3-balanced coloring c : V → {1, 2, 3} by defining c(v) = j.
Let v 1 , v 2 be distinct leaves and u 1 , u 2 be the only vertices of G adjacent to v 1 , v 2 respectively (u 1 and u 2 are not necessarily distinct). Let also G be the complete subgraph of G on the set V \{v 1 , v 2 }. Then we still have the inequality d max (G ) ≤ d max (G) ≤ k ≤ |G | 3 . Hence, by inductive assumption, there exists a strongly 3-balanced coloring c : V → {1, 2, 3} of G .
Case 3. r = 0. The major difference in this case compared with the previous two cases is that when we obtain G by deleting some arbitrary three leaves v 1 , v 2 , v 3 from G (G possesses three leaf vertices unless it is isomorphic to a string) we may loose the inequality d max (G ) ≤ |G | 3 . Also, suppose u 1 , u 2 , u 3 are the vertices adjacent to v 1 , v 2 , v 3 respectively (u 1 , u 2 , u 3 are not necessarily distinct). If we have the inequality d max (G ) ≤ |G | 3 then by inductive assumption we would have a strongly 3-balanced coloring c : V \{v 1 , v 2 , v 3 } → {1, 2, 3}, however, if c (u 1 ) = c (u 2 ) = c (u 3 ) then it becomes problematic to extend c to a strongly 3-balanced coloring c : V → {1, 2, 3}. Thus we need to employ different and more careful tactics.
We will prove the following lemma which suffices for the proof of Proposition 5.2 in the case r = 0.
Lemma 5.3. Let G = (V, E) be a tree with n = 3k vertices where d max (G) ≤ k, and let p, q be distinct pre-leaf vertices of G. Then there exists a strongly 3-balanced coloring c : V → {1, 2, 3} such that c(p) = c(q).
Proof. The proof of the lemma will be again by induction on k.
The "c(p) = c(q) part" of the claim will be needed to make the step of the induction. For k ≤ 2, the graph G is isomorphic to a string thus the claim is obvious. For k = 3 it can be seen by a direct checking (we leave this to a reader as a simple exercise). So let us assume that k ≥ 4.
Let also deg(p) ≤ deg(q). We will consider the following cases (the notations in each case will be independent of the notations of other cases): Case A: W = ∅ and p is not special.
Case B: W = ∅ and p is special.
Let v 1 be the only leaf adjacent to p, u be the unique non-leaf vertex adjacent to p, v 2 be a leaf vertex not adjacent to u, and w be the unique vertex adjacent to v 2 . We let G be the complete subgraph on V \{v 1 , v 2 , p}. Then |G | = 3(k − 1) and d max (G ) ≤ k − 1. By inductive hypothesis, there exists a strongly 3-balanced coloring c 0 : V → {1, 2, 3}. Then we extend c 0 to a strongly 3-balanced coloring c : V → {1, 2, 3} as follows: we let c(p) ∈ {1, 2, 3} such that c(p) is distinct from c 0 (u) and c 0 (q). Then we define c(v 2 ) ∈ {1, 2, 3} such that c(v 2 ) is distinct from c 0 (w) and c(p). Finally we let c(v 1 ) ∈ {1, 2, 3} such that c(v 1 ) is distinct from c(p) and c(v 2 ). Notice also that we obtain c(p) = c(q). This case is similar to Case A. Since |W | = 1 and deg(p) ≤ deg(q), we have p = v 0 . If q = v 0 , we let v 1 , v 2 , v 3 be leaves adjacent to p, q, v 0 respectively; and if q = v 0 , we let v 1 , v 2 be leaves adjacent to p, q respectively, and v 3 be a leaf not adjacent to any of the vertices p, q. We define G to be the complete subgraph on V \{v 1 , v 2 , v 3 }. Then d max (G ) ≤ |G | 3 hence G admits a strongly 3-balanced coloring c : V \{v 1 , v 2 , v 3 } → {1, 2, 3} such that c (p) = c (q). We extend c to a strongly 3-balanced coloring to c : V → {1, 2, 3} as in Case A.
Case D: |W | = 1, W = {v 0 }, p is special and there exists a leaf vertex adjacent to v 0 . This case is similar to Case B. Let v 1 be the only leaf adjacent to p, u be the unique non-leaf vertex adjacent to p, v 2 be a leaf vertex adjacent to v 0 . We let G be the complete subgraph on V \{v 1 , v 2 , p}. Then |G | = 3(k − 1) and d max (G ) ≤ k − 1. By inductive hypothesis, there exists a strongly 3-balanced coloring c 0 : V \{v 1 , v 2 , p} → {1, 2, 3}. Then we extend c 0 to a strongly 3-balanced coloring c : V → {1, 2, 3} as follows: we let c(p) ∈ {1, 2, 3} such that c(p) is distinct from c 0 (u) and c 0 (q). Then we define c(v 2 ) ∈ {1, 2, 3} such that c(v 2 ) is distinct from c 0 (v 0 ) and c(p). Finally we let c(v 1 ) ∈ {1, 2, 3} such that c(v 1 ) is distinct from c(p) and c(v 2 ).
Then, necessarily, there exists a special vertex v adjacent to v 0 . Let v 1 be the unique leaf adjacent to v. Let also v 2 be a leaf not adjacent Proposition 5.4. Let G = (V, E) be a tree with n vertices where d max (G) ≤ n k and k ≥ 3. Then G is strongly k-balanced. Proof. The proof is by induction on k. For k = 3, the claim is true by Proposition 5.2.
Assume now k ≥ 4. Then the tree G has m = n k vertices v 1 , . . . , v m such that d(v i ) ≤ 2, 1 ≤ i ≤ m, moreover, for all distinct i, j ∈ {1, . . . , m}, the vertices v i and v j are not connected by an edge. Let also V 0 = {v 1 , . . . , v m }, and G 1 be a complete graph on the subset V \V 0 . Then G 1 is a forest with n − m vertices but with d max (G 1 ) ≤ d max (G). Then G 1 is a subgraph of a tree G 2 with n − m vertices where d max (G 2 ) ≤ d max (G).
Now, for random labeled trees, Theorem B follows immediately from Theorem 4.1 and Proposition 5.4; and for random unlabeled trees, it follows immediately from Theorem 4.5 and Proposition 5.4.