Minimum congestion spanning trees in bipartite and random graphs

Acta Mathematica Scientia 2011,31B(2):634–640

http://actams.wipm.ac.cn

MINIMUM CONGESTION SPANNING TREES IN

BIPARTITE AND RANDOM GRAPHS∗

M.I. Ostrovskii

Department of Mathematics and Computer Science, St. John’s University, 8000 Utopia Parkway,

Queens, NY 11439, USA

E-mail: [email protected]

Abstract The first problem considered in this article reads: is it possible to find upper

estimates for the spanning tree congestion in bipartite graphs, which are better than those

for general graphs? It is proved that there exists a bipartite version of the known graph

with spanning tree congestion of order n3

2 , where n is the number of vertices. The second

problem is to estimate spanning tree congestion of random graphs. It is proved that the

standard model of random graphs cannot be used to find graphs whose spanning tree

congestion has order greater than n3

2 .

Key words Bipartite graph; random graph; minimum congestion spanning tree

2000 MR Subject Classification 05C05

1 Introduction

In this article, we consider finite simple graphs. Our graph-theoretic terminology follows

[1]. For a graph G, we denote its vertex set and its edge set by VG and EG, respectively.

Let G be a graph and T be a spanning tree in G (that is, T is both a tree and a spanning

subgraph of G). For each edge e of T , let Ae and Be be the vertex sets of the components of

T \e. By eG(Ae, Be) we denote the number of edges with one end vertex in Ae and the other

end vertex in Be. We define the edge congestion of G in T by

ec(G : T ) = maxe∈ET

eG(Ae, Be).

The name comes from the following analogy. Imagine that edges of G are roads, and

that edges of T are those roads which are cleaned from snow after snowstorms. For each road

g ∈ EG, there exists a unique path Pg in T joining the end vertices of g, we call such path a

detour, even in the case when Pg = g. For an edge h of T , it is quite natural to define the

congestion c(h) as the number of times h is used in different detours {Pg}g∈EG. Then,

ec(G : T ) = maxh∈ET

c(h).

∗Received September 15, 2007; revised March 9, 2009.

No.2 Ostrovskii: MINIMUM CONGESTION SPANNING TREES 635

It is clear that, for applications, it is interesting to find a spanning tree minimizing the conges-

tion.

We define the spanning tree congestion of G by

s(G) = min{ec(G : T ) : T is a spanning tree of G}. (1)

Each spanning tree T in G satisfying ec(G : T ) = s(G) is called a minimum congestion span-

ning tree for G. The parameters ec(G : T ) and s(G) were introduced and studied in [2]. These

parameters are of interest in the study of Banach-space-theoretical properties of Sobolev spaces

on graphs, see [3, Section 3.5.1]. An additional motivation for this study is that minimum con-

gestion spanning trees can be considered as ‘congestional’ analogues of the well-known shortest

(or minimal) spanning trees. See [4] for an account on shortest spanning trees, and on other

minimality properties of spanning trees.

One of the natural problems about s(G) is to find the maximal possible value μ(n) of s(G)

for graphs G with n vertices. The rate of growth of the function μ(n) was studied in Section

3 of [2], where it was shown that c1n3

2 ≤ μ(n) ≤ c2n2 for some absolute constants c1 and c2,

and some estimates for c1 and c2 were found. The first problem considered in this article is:

does the lower bound of μ(n) still hold in the case of bipartite graphs? In connection with

this problem, we prove that there exists a bipartite version of the graph with spanning tree

congestion of order n3

2 from [2, Theorem 2], but it has to have a radius larger than that in the

example from [2]. In Section 3, we study spanning tree congestion of random graphs. As is well

known, random graphs can be used to find examples with close-to-extremal behavior for many

problems. We prove that the standard model of random graphs cannot be used to improve the

exponent 32 in the lower estimate for μ(n).

2 Minimum Congestion Spanning Trees in Bipartite Graphs

In [2, Theorem 2], it was shown that there exist graphs G with n vertices and s(G) ≥ cn3

2 ,

where c > 0 is an absolute constant. Our first purpose is to show that (with a bit worse

estimate for c) the same result remains true for bipartite graphs. We refer to [5] for a systematic

exposition of the theory of bipartite graphs.

We use the following way for getting a bipartite graph from an arbitrary graph, and we

call it duplication. Let H be a graph with vertex set VH . By duplication of the graph H we

mean the bipartite graph H defined in the following way. We replace each vertex v ∈ VH by

a pair of vertices uv, wv, and let VH = {uv, wv : v ∈ VH}. The vertex uv is adjacent to wx in

H if and only if x ∈ VH is either v or a vertex adjacent to v. It is clear that H is a bipartite

graph and that the sets {uv}v∈VHand {wv}v∈VH

form a bipartition.

We are going to show that the spanning tree congestion of the duplication of the graph G

considered in [2, Theorem 2] satisfies the same estimate below as s(G). Recall the definition of

G:

Let n be of the form n = 3k− 2√k, where k is such that√

k is an integer. We also assume

that k > 4. The graph G has n vertices and is defined in the following way. We represent VG as

a union of 3 sets: VG = C1∪C2∪C3, where |C1| = |C2| = |C3| = k, |C1∩C2| = |C2∩C3| =√

k,

636 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B

and C1 ∩ C3 = ∅. The edge set of G is defined in the following way: vertices v and u in G are

adjacent if and only if u, v ∈ Ci for some i ∈ {1, 2, 3}.Theorem 1 Denote the duplication of the graph G by G. Then, G is a bipartite graph

with 2n = 6k − 4√

k, k > 4, vertices and s(G) ≥ 12k3/2.

Remark Our proof follows the lines of the proof of Theorem 2 in [2], with modifications

needed because of the duplication.

Proof Working with trees, it will be convenient to use related definitions and observations

that are going back to C. Jordan [6], see [7, pp. 35–36]. Let u be a vertex of a tree T . If we

delete all edges incident with u from T , we get a forest. The maximal number of vertices in

components of the forest is called the weight of T at u. A vertex v of T is called a centroid

vertex if the weight of T at v is minimal. Each tree has one or two centroid vertices.

Let T be a spanning tree in G satisfying s(G) = ec(G : T ), and c be a centroid of T .

If T has two centroids, we take one of them and fix the notation c for it. Denote the sets

{wv}v∈Ciand {uv}v∈Ci

by Di and Ei, respectively, and let Ci = Di ∪ Ei. As the sets C1

and C3 are disjoint, the centroid c does not belong to at least one of them. Without loss of

generality, we assume c /∈ C1. The edges incident with c in T will be called central edges. If

we remove all central edges from T , we get a forest. Let V1, · · · , Vt be vertex sets of connected

components of the forest, except the component whose only vertex is c. There is a natural

bijective correspondence between the set of central edges and the set {V1, · · · , Vt}. Let us

denote the central edge corresponding to Vi by ei.

Observe that C1 cannot intersect more than 2√

k of the sets V1, · · · , Vt. In fact, only 2√

k

vertices of C1 are adjacent to vertices that are not in C1, so there are only 2√

k “entrances”

into C1.

As C1 contains 2k vertices, this implies that there exists j ∈ {1, · · · , t} such that the

intersection Vj ∩ C1 has at least√

k vertices.

Let γ1 = |Vj ∩ C1|. Then, γ1 = δ1 + ε1, where δ1 = |Vj ∩ D1| and ε1 = |Vj ∩ E1|.There are δ1(k − ε1) + ε1(k − δ1) = γ1k − 2ε1δ1 edges joining Vj ∩ C1 with C1\Vj . Also,

ε1δ1 ≤ γ2

1

4 . Hence, ej is used in at least(k − γ1

2

)γ1 detours. As γ1 ≥

√k and k > 4, then,

either(k − γ1

2

)γ1 ≥ 1

2k3/2 or k − γ1

2 < 12

√k. In the former case, we are done. In the latter

case, γ1 = ε1 + δ1 > 2k −√k, hence, at most√

k of the 2√

k elements of the set C1 ∩ C2 are

not in Vj , and we get |C1 ∩ C2 ∩ Vj | >√

k. Therefore, |C2 ∩ Vj | >√

k. Let γ2 = |C2 ∩ Vj |.By arguing in the same way as above, there are more than

(k − γ2

2

)γ2 edges joining Vj∩C2

with C2\Vj . Hence, ej is used in at least(k − γ2

2

)γ2 detours. As γ2 >

√k and k > 4, then,

either(k − γ2

2

)γ2 ≥ 1

2k3/2 or k − γ2

2 < 12

√k.

In the former case, we are done. In the latter case, we obtain

|Vj | > |C1|+ |C2| − |C1 ∩ C2| − |C1\Vj | − |C2\Vj | = 4k − 4√

k.

As k > 4, this inequality implies |Vj | > |VG|/2. We get a contradiction with the fact that c is

a centroid.

It would be interesting for general graphs: How does the duplication affect the parameter

s(G)? In particular, we suggest the following problem:

Problem 1 Let G be a connected graph and G be the graph obtained by duplication

of G. Is it true that s(G) ≥ s(G)? (Or, at least, s(G) ≥ c · s(G) for some absolute constant


c > 0?)

The stated problem is a special case of the following more general problem: How do

standard unary operations (such as complementation or line graph) affect the parameter s(G)?

Recall (see [1, p. 10]) that the radius of a graph G is minvmax

wdG(v, w), where dG is the

standard metric on G. Observe that the graph from [2, Theorem 2] described above has radius

2, and its duplication has radius 3. The following result shows that, for bipartite graphs of

radius 2, the quantity s(G) cannot be large.

Theorem 2 Let G be a bipartite graph of radius 2. Then, s(G) ≤ 3|VG|.Proof Consider a vertex v in G satisfying dG(u, v) ≤ 2 for every u ∈ VG.

Let B = {u ∈ VG : dG(u, v) = 1}. Let m = |B|, where B = {b1, · · · , bm}. Let A = {u ∈VG : dG(u, v) = 2}. We consider a partition {Ai}m

i=1 of A satisfying the following conditions:

• All vertices from Ai are adjacent to bi.

• The summ∑

i,j=1

||Ai| − |Aj || has the minimal possible value.Let T be the tree with the following edge set

ET = {vbi : i = 1, · · · , m} ∪ {bia : a ∈ Ai, i = 1, · · · , m}.

(By uv we denote the edge joining vertices u and v.) Clearly, T is a spanning tree and an edge

bia is used in da(≤ n− 1) detours, where da is the degree of a.

So, it remains to estimate the number of detours using an edge of the form vbi. If U and

W are sets of vertices of G, we use the notation EG(U, W ) for the set of edges with one end

vertex in U and the other end vertex in W . The edge vbi is used in detours for

D := EG

({bi} ∪Ai, {v} ∪ {bj}j �=i

⋃( ⋃j �=i

Aj

))

= {biv}⋃

EG({bi},⋃j �=i

Aj)⋃

EG(Ai, {bj}j �=i).

As G is bipartite, there is no edge between any pair of vertices in A (B). Let n = |VG|. Then,

|D| ≤ 1 + (n− (m+ 1)) + |EG(Ai, {bj}j �=i)|.

Observation A vertex bt is not adjacent to any vertice from As if As satisfies |As| >

|At|+ 1.

In fact, otherwise, we take a vertex from As that is adjacent to bt, and move it to At.

We get a new partition {A′k} satisfying |A′s| = |As| − 1, |A′t| = |At| + 1, and |A′k| = |Ak| fork /∈ {s, t}.

It is seen thatm∑

i,j=1

∣∣|A′i| − |A′j |∣∣ <

m∑i,j=1

||Ai| − |Aj || .

By the observation,

|EG(Ai, {bj}j �=i)| ≤ |Ai| · |{j : |Aj | ≥ |Ai| − 1}| ≤ |Ai|n− (m+ 1)

|Ai| − 1≤ 2(n− (m+ 1)),

if |Ai| > 1. We obtain an estimate |EG(Ai, {bj}j �=i)| ≤ m − 1 if |Ai| = 1. Hence, |D| ≤max{3n− 3m− 2, n− 1}. Therefore, ec(G : T ) < 3n = 3|VG|.


3 Minimum Congestion Spanning Trees in Random Graphs

Let G(n, p) be the standard probability space of graphs (see [1, p. 217]). We denote the

standard probability on this space by Pp and its random element by Gn,p. Our purpose is to

show that the spanning tree congestion of graphs in G(n, p) is, with high probability, O(n3

2 ).

Our approaches showing that the probability of the event “the spanning tree congestion is

small” is large are different in the cases p ≤ n1

2 and p ≥ n1

2 . In the first case, the random graph

just does not have enough edges to have ‘large’ congestion. In the second case, the graph has,

with high probability, nice spanning trees (see Theorem 3 for precise meaning of this statement),

whose existence implies the desired bound for the congestion.

First, we consider the case of ‘small’ p. Let A = An,p be the intersection of the following

two events in G(n, p): {Gn,p is connected} and {Gn,p : s(Gn,p) ≥ n3

2 }.Proposition 1 If p ≤ n−

1

2 , then, Pp(A) ≤ exp(− 3

16n3

2

).

Proof We use the following straightforward estimate s(G) ≤ |EG| − |VG| + 2 (see [2,

Theorem 1(a)]). Hence, Pp(A) ≤ Pp{Gn,p has at least n3

2 edges}. The expected number λ of

edges of Gn,p isn(n−1)

2 p. If p ≤ n−1

2 , then, λ < n3

2

2 . By a Chernoff-type bound (see [8, Formula

(2.5), p. 26]),

Pp{Gn,p has at least n3

2 edges} ≤ exp

⎛⎝− (n

3

2 − λ)2

2(λ+ n

3

2−λ3

)⎞⎠

≤ exp

(− (n

3

2 /2)2

2 · ( 23n

3

2

))= exp

(− 3

16n

3

2

).

Now, let n−1

2 ≤ p ≤ 1 and let n ∈ N be fixed (the discussion below assumes, also, that n is

not very small). Let s = 12p(n− 1)�, and d =

⌈log2

(n−1

s

)⌉. We assume that n is large enough

so that s ≥ 2. A tree T will be called a nice tree of depth k, k ∈ {0, 1, 2, · · · , d} correspondingto the pair (p, n) if it has a vertex v0 of degree s, at most n vertices, and the cardinalities of

the vertex sets of the s components of T obtained after removal of v0, we call such components

branches, satisfy the following conditions:

• If 0 ≤ k ≤ ⌊log2

(n−1

s

)⌋, then each of the branches has 2k vertices.

• If log2

(n−1

s

)is not an integer, and k =

⌈log2

(n−1

s

)⌉, then, each of the branches has

vertices between 2k−1 and 2k in number, and the tree has n vertices.

Denote the event of existence of a subgraph of Gn,p which is a nice tree of depth k by

Tk ∈ G(n, p).

Theorem 3 For each k ∈ {0, · · · , d}, the estimate Pp(Tk) ≥ (1 − exp(−c1√

n))k+1 ≥1− exp(−c2

√n) holds, where c1 > 0 and c2 > 0 are absolute constants.

Remark To relate this theorem with our study of minimum congestion spanning trees,

we observe that a nice tree of depth d has the following properties:

• It is a spanning tree.

• If we delete any of its edges, one of the obtained components has at most 2d ≤ 2 · n−1s

vertices. Hence, the edge is used in at most 2n · n−1s detours.

Therefore, these conditions imply (for each graph in which T is a spanning tree) that

ec(G : T ) ≤ 2n · n−1s ≤ Cn

3

2 .


Proof of Theorem 3 We estimate Pp(Tk) by induction. Observe that T0 is the event

of existence of a vertex whose degree is at least s. The high probability of this event follows

from a Chernoff-type estimate. In fact, the expected degree of a vertex is p(n− 1). Hence, for

a fixed vertex v0 (whose degree we denote by dv0), the probability α := Pp{dv0

≤ 12p(n − 1)}

can be estimated using [8, inequality (2.6), p. 26]. We obtain

α ≤ exp

(− (

12p(n− 1))2

2p(n− 1)

)= exp

(−18p(n− 1)

)≤ exp

(−n− 1

8√

n

)

and

Pp(T0) ≥ 1− exp

(−n− 1

8√

n

)≥ 1− exp(−c1

√n).

To estimate Pp(Tk) for k > 0, we use conditional probabilities: Pp(Tk) = Pp(Tk−1) ·Pp(Tk|Tk−1). We estimate Pp(Tk|Tk−1) using the following lemma. Recall that a set of edges

is called a matching if no two of them have a common vertex.

Lemma 1 Let A and B be disjoint vertex sets satisfying |A| ≤ n2 � and |B| ≥ n

2 �. Then,the probability that Gn,p does not contain a matching FA, such that each edge from FA has

one end vertex in A and one end vertex in B, and each vertex from A is incident with exactly

one edge from FA, is ≤ exp(−c1√

n) for some absolute constant c1 > 0.

Proof It is clear that the probability is the least if |A| = |B| = n2 �. In such case, the

probability can be estimated using the method from B. Bollobas and A. Thomason [9, pp. 63–

65] (see, also, [10, pp. 158–159] and [8, pp. 82–84]). As in our case p ≥ n−1

2 , we can use more

crude and simple estimates. We denote n2 � by m and write the estimates from [9] with m

instead of n. We need a version of the following lemma from [9, Lemma 6, p. 64]. As mentioned

in [9], it can be derived from Hall’s theorem (which can be found, e.g, in [1, p. 77]).

Lemma 2 [9] Let G be a bipartite graph with vertex classes V1 and V2, |V1| = |V2| = m.

Suppose that G does not have a matching F , such that each of the vertices is incident with

exactly one edge from F . Then, there is a subset N ⊂ Vi (i = 1, 2) such that

(i) Γ(N) := {y : y is adjacent to some x ∈ N} has |N | − 1 elements.

(ii) 1 ≤ |N | ≤ (m+ 1)/2.

We consider the bipartite subgraph of Gn,p whose vertex classes are V1 = A and V2 = B

(the edge set of this subgraph consists of all edges of Gn,p having one end vertex in A and one

end vertex in B). Denote by Ia (a = 1, 2, · · · , (m+1)/2�) the event that there is a set N ⊂ Vi

(i = 1, 2), |N | = a satisfying (i) and (ii) for this subgraph. As we have 2

(m

a

)choices for N

with |N | = a and

(m

a− 1

)more choices for Γ(N), we obtain

Pp(Ia) ≤ 2

(m

a

)(m

a− 1

)(1− p)a(m+1−a).

Using the elementary estimates

(m

a

)≤ ma and 1− p < e−p, we obtain

Pp(Ia) ≤ 2m2a exp(−pa(m+ 1− a)) ≤ 2

(m2 exp

(−p · m+ 1

2

))a

.


Let m1 =⌊

m+12

⌋. By Lemma 2, the probability of non-existence of a set FA satisfying the

condition of Lemma 1 can be estimated from the above by Pp

( m1⋃a=1

Ia

). We have the estimate

Pp

( m1⋃a=1

Ia

)≤

m1∑a=1

2

(m2 exp

(−p · m+ 1

2

))a

≤ 2m2 exp(−p · m+1

2 )

1−m2 exp(−p · m+12 )

≤ exp(−c1n1

2 )

for some absolute constant c1 and sufficiently large n. (Of course, we can select the same

constant here and in our estimate for Pp(T0).)

We continue our proof of the theorem. Suppose that we have proved the estimate for Tk−1.

If a nice tree of depth k − 1 has n vertices, there is nothing more to prove. Otherwise, it has

2k−1s + 1 vertices. Observe that for each set of 2k−1s + 1 vertices in Gn,p the existence of a

nice tree with this vertex set is independent from the existence of edges between vertices from

this set and the remaining vertices. Let T be a nice tree of depth k − 1.

We consider two cases.

Case 1. 2ks + 1 ≤ n. In this case, there are at least 2k−1s vertices in Gn,p that are

not vertices of T . Let A be the union of the vertex sets of all branches of T and B be the

complement of the vertex set of T . If we add to T a matching FA satisfying the conditions of

Lemma 1 (and its vertices), we obtain a nice tree of depth k, hence,

Pp(Tk|Tk−1) ≥ Pp(existence of FA|Tk−1) = Pp(existence of FA) ≥ 1− exp(−c1n1

2 )

(we used the independence mentioned above). Hence, Pp(Tk) ≥ Pp(Tk−1)(1 − exp(−c1n1

2 )) in

this case.

Case 2. 2ks+ 1 > n. In this case, k = d, and the number of vertices in Gn,p that are not

in T is < 2k−1s. We denote this set by A and let B the set of all vertices of T except v0. We

apply Lemma 1 to these A and B, and denote the corresponding matching by FA. Adding FA

(and its vertex set) to T , we obtain a nice tree of depth d. The probability is estimated in the

same way as in Case 1.

Problem 2 It would be interesting to find more precise estimates for the spanning tree

congestion of random graphs. It seems plausible that “most” random graphs in G(n, p) satisfy

s(Gn,p) ≤ C · n for some absolute constant C.

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[5] Asratian A S, Denley T M J, Haggkvist P. Bipartite graphs and their applications. Cambridge: Cambridge

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[6] Jordan C. Sur les assemblages de lignes. J Reine Angew Math, 1869, 70: 185–190

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Minimum congestion spanning trees in bipartite and random graphs