uniquely k -arborable graphs

UNIQUELY k-ARBORABLE GRAPHS

BY JOHN MITCHEM

ABSTRACT

A graph is uniquely k-arborable if its point-arboricity is k and there is only one acyclic partition of its point set into k subsets. Several properties of uniquely k-arborable graphs are presented. One such property is that uniquely k-arborable graphs are (k -- 1)-cormected. Furthermore, it is shown that for any positive integer k there is a uniquely k-arborable graph which is not k-connected.

Introduction

For any set S of points of a graph G the subgraph (S) induced by S is the

maximal subgraph of G with point set S. A partition of the point set of a graph is

said to have property P if each set in the partition induces a graph with property P.

A graph is totally disconnected if it contains no lines. Thus the chromatic number

z(G), of a graph G may be defined as the minimum number of subsets in any

totally disconnected partition of the point set of G. If z(G) = k and there is only

one totally disconnected partition of the point set of G into k sets, then G is said

to be uniquely k-colorable. The properties of uniquely k-colorable graphs have

been investigated in [1], [2], and [6].

Analogous to chromatic number is the concept of point-arboricity of a graph G,

denoted p(G), which is defined as the minimum number of subsets in any acyclic

partition of the point set of G. A graph G is said to be uniquely k-arborable if

p(G) = k and there is only one acyclic partition of the point set of G into k sets.

Point-arboricity has been studied in [3], [4], and [5]. In this paper we developed

both necessary and sufficient conditions for a graph to be uniquely k-arborable.

Specially we show that:

1) For any positive integer k there is an infinite class of uniquely k-arborable

graphs.

Received November 17, 1970

17

18 JOHN MITCHEM Israel J. Math.,

2) Every uniquely k-arborable graph is (k - 1)-connected, and that for k > 2

there is a uniquely k-arborable graph which is not k-connected.

3) No planar graph is uniquely k-arborable for k > 3 and no outerplanar

graph is uniquely k-arborable for k > 2.

Sufficient conditions

We begin with some notation and preliminary observations.

The minimum degree of graph G is denoted by 6(G). We note that every acyclic

graph is uniquely 1-arborable. Also, let 111,--', Fk be the acyclic partition of a

uniquely k-arborable graph G. Then for i = 1,.-., k each point v in V1 is adjacent

to at least two points of Vj, j ~ i, because otherwise, adding v to set Vj would

form a second partition of V(G) into k or fewer acyclic sets. This implies that

6(G) > 2k - 2 for any uniquely k-arborable graph G, and each V~, i = 1, ..., k has

at least two points.

We now show that in verifying that a graph G is uniquely k-arborable, it is

sufficient to show that the point set of G, which is denoted by V(G), has a unique

acyclic partition into k sets.

THEOREM 1. Let G be a graph of order p and let 1 < k < p. Then G is

uniquely k-arborable if and only if there is a unique acyclic partition of V(G)

into k sets.

PROOF. The necessity is immediate from the definition of uniquely k-arborable

graphs, so we consider only the sufficiency.

Suppose there is a unique acyclic partition of V(G) into k sets. This implies

that p(G) < k. Assume p(G) = m < k, and let VD'", Vm be an acyclic partition of

V(G). At most one of the Vi, say V,,, has less than two points. We now form an

acyclic partition of V(G) into k sets. Form singleton subsets of 111 until either V~

has been completely partitioned into sets with one element or the sets V~, i > 1,

together with the sets in the partition of V 1 form exactly k sets. If the former

occurs, we partition V2 into single-element subsets until either V2 has been comple-

tely partitioned into sets with one element or the partitions of V1 and 1/2 together

with sets V3,-'-, Vr, form a partition of V(G) into k sets. Continuing in this way we

obtain an acyclic partition of V(G) into sets W~, W2,--', Wk. Since m < k, at least

one set, say W,, contains only a single element. Also, k < p implies that at least

one set, say W 2, contains at least two elements.

From the partition W~,..., Wk, we form another acyclic partition by adding v,

Vol. 10, 1971 UNIQUELY k-ARBORABLE GRAPHS 19

an element of W2, to set 1411. Thus we have two distinct acyclic partitions of V(G)

into k sets. This contradiction implies that p(G) = k, which completes the proot:

The union of two graphs Gx and G z is the graph whose line set is the union of

the line sets of G1 and G2 and whose point set is V(G1) td V(G2). Two graphs are

said to be disjoint if their point sets are disjoint. Let Ga, . . . ,G, be mutually

disjoint graphs. The join of G 1 and G 2, denoted by G 1 + G z = ]~lZGi, is the graph

which consists of G1 u G2 together with all lines joining V(GO with V(Gz).

For n > 3, the graph Y~Gi is defined as (]E~'-IG~) + (7,. A star is a graph of the

form Ha + H2 where H~ consists of a single point and Hz is any totally discon-

nected graph.

The next theorem exhibits an infinite class of uniquely 2-arborable graphs. We

shall then employ this result to obtain a sufficient condition for a graph to be uniquely k-arborable, k > 2.

THEOREM 2. Let G1 and G 2 be disjoint acyclic graphs having at least three

points and one line. I f the graph G 2 with its isolated points deleted is not a star,

then G = G 1 + G 2 is uniquely 2-arborable.

PROOF. The point-arboricity of G is two and V(GO, V(G2) is an acyclic partition

of G. Assume Wa, WE is another acyclic partition of V(G). Then it is easily seen

that one of W a and W2 contains all but one point of V(G 0 and the other contains

all but one point of V(G2). Thus we suppose, without loss of generality, that for

i = 1, 2, Wi contains exactly one point vi of V(Gi). That is, W 2 : (V(GI) U {v2}) - {vl} and W~ = (V(G2) k.) {va} ) - - { V 2 } o However

G2 does not have all its lines incident with a single point so there are points v and

w both different from v2 which are adjacent in G2. Thus the set {vl, v, w} induces

a triangle in (Wa) which contradicts the fact that (1411) is acyclic.

THEOREM 3. For k > 1, let G i = (V~,Ei), i = 1,2, . . . ,k , be mutually disjoint

acyclic graphs with at least 3 points and one line. If, for i = 2 ,3 , . . . ,k , the

deletion of the isolated points of Gi does not result in a star, then G = y.RG i is

uniquely k-arborable.

PROOF. We use induction on k. Theorem 2 establishes the result for k = 2.

Assume that for k > 3 any graph which is the join of k - 1 acyclic graphs having

the properties given in the hypothesis of the theorem is uniquely (k - 1)-arborable.

Then Hi = G - Gi is uniquely (k - 1)-arborable for i = 1,2, . . . ,k.

We assume that there are two acyclic partitions of V(G) into/¢ sets and show

that this leads to a contradiction. One acyclic partition of V(G) is V1, "' , Vk, and


let W~, ..., Wk be another. Now, we verify three remarks on the relationship

between the sets in each partition and use these observations to complete the

proof of the theorem.

REMARK A. For every pair of sets Vi, W i, the set V i is not contained in

and W~ is not contained in V s. In order to prove this, we assume the contrary and

consider two cases.

Case i. There exists integers s and r such that V~ is contained in W,. If

V~ -- W, then {Wi}i~, is a second acyclic partition of Hs, which contradicts our

inductive assumption. In the case where V, # W~ there is a point v in W, - V~.

Since (Vs> contains a line, < Wr> has a cycle, which is impossible.

Case ii. There exist integers s and r such that W, is contained in but not

equal to V,. In this case, if {W/n Hs}i~ r forms a second acyclic partition of H~, we

have a contradiction to the inductive assumption.

Otherwise, {W/NH~}i~ r = {Vj)j,~ which implies some set Vj is contained in

some Wi. However, Case i shows that is impossible.

In both cases we have a contradiction, and thus Remark A is verified. Each set

W~ therefore contains points from two different Vj, and each set Vj contains points

from two distinct W,. Clearly no set W~ has points from three different Vj and no

141/contains two points from each of two distinct Vj. Thus each W i contains ex-

actly one point from one Vj and all of its other points are from another Vj.

REMARK B. No set Vj has points in three different Vgi. In order to establish

this, we assume that V,, for some integer r, has points in three different W~. For

j ~ r, Remark A implies Vj has points in at least two different 14/,. Thus there are

at least 2k + 1 non-empty intersections of the sets Vj with the W~ as i ,j take on

all distinct values 1,2, ..., k. However, each ~ intersects exactly two Vj so that

there are precisely 2k non-empty intersections of the Vj with the ~ . This con-

tradiction proves Remark B.

REMARK C. Each Wi consists of exactly one point from one Vj and all but

one point of another V s. If this remark is not valid, then there is a set Wi which

must contain two points from each of two different V s. We have already seen that

this is impossible and so Remark C is proved.

We now select r such that 1/1 n Wr contains a single point v. Then there is an

integer s ~ 1 such that w ~ V~ and W, = {v} u (V~ - {w}). By hypothesis, not all

lines of G~ are incident with a single point. Thus <Vs - {w}> has a line and <W,>


contains a triangle. This contradiction implies there is a unique acyclic partition

of V(G) into k sets. So by Theorem 1, G is uniquely k-arborable which completes

the proof.

Necessary conditions

We now present a number of necessary conditions for uniquely k-arborable

graphs. It is easily seen that, for k > 1, any uniquely k-arborable graph is con-

nected. In order to give a stronger result, we state the following.

THEOREM 4. In any uniquely k-arborable graph, k > 1, the union of any

two (of the k) sets of the acyclic partition induces a connected graph which

contains a cycle.

THEOREM 5. / f graph G is uniquely k-arborable, k > 2, then G is (k - 1)-

connected.

PROOF. I f k = 2, the desired result is implied by the aforementioned observation

that G is connected. Thus, let k > 2, and suppose G is not (k - 1)-connected.

Then there is a set S of k - 2 points whose removal disconnects G. There are

two sets V 1 and V 2 of the acyclic partition of V(G) into k sets which have no points

in S. According to Theorem 4, the union of V1 and V2 induces a connected graph.

That is, all points of VI and V2 are in the same component of G - S. Another

acyclic partition of V(G) into k sets can be obtained by adding any point v in

another component of G - S to set V 1. This contradicts the fact that G is uniquely

k-arborable and proves Theorem 5.

Since, for 2 _< m ___ k, the union of any rn of the k acyclic sets of a uniquely

k-arborable graph induces a uniquely m-arborable graph, we have the following

result.

COROLLARY 5a. Let G be a uniquely k-arborable graph with acyclic partition

V1, . . . , V k. For 2 < m < k, the union of any m sets of the partition induces an

(m - 1)-connected graph.

The next theorem shows that the conclusion of Theorem 5 cannot be improved.

THEOREM 6. For every k > 2, there is a uniquely k-arborable graph which

is not k-connected.

PROOF. For each k > 2, we define a graph G (k) which has a cut set of k - 1

points and is uniquely k-arborable.

22 J O H N M I T C H E M Israel J. Ma th . ,

Let Gt, G2, "", Gk denote mutually disjoint graphs such that, for i = 1, 2, ..., k - 1,

G~ is the path v1"(°,~2" (0, ..., v(70 and Gk consists of two disjoint paths

P = v(lk),v(2k),v(a k) and P ' = v(4k),v(sk),v (k).

For i = 1 , 2 , . . . , k - 1, let

L (~) = ;rv(!)" j = 1 2, 3, 4} t j • , and

R (° = {v~°:j = 4, 5, 6, 7}.

Also, let

Further, define

L (k) = {v~k): j = 1,2, 3}

R (k) = {vSk':j = 4,5,6}.

k k

L = [ , . J L (0and R = [..J R (0. 1 1

and

Now denote by G (k) the graph which consists of the union of the k mutually

disjoint graphs G~ together with all possible lines joining points in distinct L(0

and all possible lines joining points in distinct R (~). The graph G ~2> is shown in

Fig. 1. Furthermore, if S is a subset of V(G(k)), then we call S n L and S n R the

left side of S and right side of S respectively.

Fig. 1

We first consider the graph G (2). Since the removal of vO~ disconnects the

graph, GcZ~ is not 2-connected. Clearly p(G) = 2, so that it remains to show that

that the only acyclic partition of V(G ¢a~) is V(G1), V(G2).

Suppose W 1, W2 is another acyclic partition of V(Gc2~). Then one of the sets,

say W1, has at least seven points and neither W1 nor W2 contains all of V(Gj) for

j = 1 or 2. Thus W1 contains points from both V(Gt) and V(G2).


Since W~ has at least seven points, one side of W~ has at least four points. If

these four points are in both V(G1) and V(G2), they induce a cycle which con-

tradicts that (W1) is acyclic. Thus we suppose all four of these points are in

V(G1). Then since v(41) is in both sets L and R, the other side of W~ must have

four points not all in V(G~). These four points induce a cycle, which again con-

tradicts that (1471) is acyclic. Hence G c2) is uniquely 2-arborable.

Induction on k is now used to show that, for k > 2, G(k)is uniquely k-arborable.

Assume G (k-l) is uniquely ( k - 1)-arborable and consider G ck~.

The sets V(GO, ..., V(Gk) form an acyclic partition of V(G (°) into k sets.

According to Theorem 1 it suffices to show that this partition is unique. Suppose

there is an acyclic partition W1,..., W k of V(G (k)) which is different from

v (ao , ..., v(c~).

Assume that one of these sets, say IV/, is equal to V(Gi) for somej = 1, 2 , . . . , k - 1.

Then W1,...,Wi_l, W~+I,...,W k is an acyclic partition of G(k)-Gj= G (k-l) different from

V(G0, ..., V(Gj_ i), V(G~+ 0 , ' " , V(Ok).

However, by the inductive assumption this is impossible.

Thus one set, call it 1411, must contain m __> 7 points and W1 ~ V(Gj) for

j = 1,--., k. These m points must come from at least two different V(Gj). But

m __> 7 implies that one side of Wx contains at least four points. Furthermore,

using the argument given for G(2), we can say that one side of W1 contains four

points in two or more different V(Gj). These four points induce a cycle which

implies that (WI) is not an acyclic graph. This contradiction implies the partition

V(GO,..., V(Gk) is unique and thus G(k~is uniquely k-arborable.

The graph G (k)- {v(~): i = 1 , - . - , k - 1} is disconnected, which implies the

connectivity of G (k)does not exceed k - 1. This completes the proof.

THEOREM 7. I f G is a uniquely k-arborable graph, k >__ 2, and V1, "", Vk is

the acyclic partition of V(G) such that I = Iv21 =< ... =< I vkl, then I vll = 3

and IV21 4

PROOF. Suppose I/i contains only two points u and v. Since (V2) is acyclic, Vz can be partitioned into sets Sx and $2 each of which induces a graph with no lines.

Hence, Sx u {v} and $2 td {u} both induce acyclic graphs which contradicts the

fact that (V1 u V2) is uniquely 2-arborable.


To verify that V2 contains at least four points we assume the contrary and let

V2 = {vl,v2,v3} and V 1 -- {ul,u2,u3}. Then two points of V 1, say u 1 and u2, are

non-adjacent as are two points, say v~ and v2, of V2. The sets {ul,uE, V3} and

{vl,v2, Ha} induce acyclic graphs and we have two different acyclic partitions of

V 1 u V 2 into two sets. This is impossible, and thus t V21 > 3.

THEOREM 8. Let G be a uniquely k-arborable graph where k > 2. I f v is a

point of G with deg v <~ 2k - 1, then G - v is uniquely k-arborable.

PROOF. Let V~,-.., Vk be the partition of V(G) into k acyclic sets, and suppose

v ~ V1. The set V 1 - v is non-empty, and V1 - v, V2, '" , Vk is an acyclic partition

of G - v. Assume W1, W2,---, Big is also an acyclic partition of G - v. Since

dega v __< 2k - 1, there is a set W~, say W~, which has at most one point adjacent

in G to v. Thus W[ = {v} t3 W1, W2,'- ' , Wk is an acyclic partition of G. There is

only one acyclic partition of V(G) into k sets. Thus W~ = V1 and without loss of

generality we may say W~ = V/, i = 2, 3,.--, k. Then W 1 = V 1 - v, WE = V2, "", IVk

= V k is the only acyclic partition of V(G - v) into k sets. Hence, according to

Theorem 1, G - v is uniquely k-arborable.

THEOREM 9. Let G O be a uniquely k-arborable graph, k ~ 2, with p points

and q lines. Then p > 4 k - 1 and q > kp.

PROOF. Theorem 7 implies that p __> 4k - 1.

I f 6(Go) > 2k, then q > kp. Thus we need only consider the case where there

is a point v o of G O such that deg Vo _-< 2k - 1. Theorem 8 implies that G~ = Go - Vo

is uniquely k-arborable. Either the graph G 1 has a minimum degree at least 2k or

there is a point vl in G~ such that degG, Vx < 2k - 1. In the former case we stop

deleting points of G. In the latter case G2 = G~ - v~ is uniquely k-arborable. I f

6(G2) > 2k, we stop deleting points of G; otherwise, there exists a point v2 e V(G2)

such that G3 = G2 - v2 is uniquely k-arborable. Theorem 8 implies that we can

continue deleting points until we obtain a graph G,, which is uniquely k-arborable,

has minimum degree at least 2k, and has been obtained by subtracting m points,

Vo, "",vm_~, f rom Go. The number of lines in G,, is at least k(p - m). We recall

that uniquely k-arborable graphs have minimum degree at least 2 k - 2. For

i = 0, 1,.. . , m - 1, G~ is uniquely k-arborable, and thus each v~ has degree in G~

not less than 2k - 2. In the removal of each point, we also removed at least

2k - 2 lines which implies that (2k - 2)m > km lines were deleted f rom Go in

obtaining G,,. Thus G o has not less than k(p - m) + k m = kp lines.


The above theorem implies that, for k > 3, any uniquely k-arborable graph

with p points has at least 3p lines. Since every planar graph of order p( > 3) has

at most 3 p - 6 lines, we arrive at the following:

COROLLARY 9a. I f G is uniquely k-arborable, k > 3, then G is non-planar.

An outerplanar graph of order p ( > 2) has at most 2 p - 3 lines, which enables us

to state another corollary.

COROLLARY 9b. For k > 2, no uniquely k-arborable graph is outerplanar.

According to Corollary 9b, no uniquely 2-arborable graph is outerplanar. The

author has been unable, however, to find a planar, uniquely 2-arborable graph.

This leads us to the following:

CONJECTURE. I f G is uniquely 2-arborable, then G is non-planar.

REFERENCES

1. D. Cartwright and F. Harary, On colorings of signed graphs, Elem. Math. 23 (1968), 85-89. 2. G. Chartrand and D. Geller, On uniquely colorable planar graphs, J. Combinatorial

Theory 6 (1969), 271-278. 3. G. Chartrand, D. Geller and S. Hedetniemi, Graphs with forbidden subgraphs, J. Com-

binatorial Theory, to appear. 4. G. Chartrand and H. V. Kronk, The point-arboricity of planar graphs, J. London Math.

Soe. 44 (1969), 612-616. 5. G. Chartrand, H. V. Kronk and C. E. Wall, Thepoint-arboricity ofagraph, Israel J. Math.

6 (1968), 169-175. 6. F. Harary, S. Hedetniemi and R. Robinson, Uniquely colorable graphs, J. Combinatorial

Theory 6 (1969), 264-271.

SAN JOSE STATE COLLEGE, SAN JOSE, CALIFORNIA

uniquely k -arborable graphs

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