Dividing a graph by degrees

Dividing a Graph by Degrees

R. P. Anstee* DEPARTMENT OF MATHEMATICS, UNIVERSITY OF BRITISH COLUMBIA

VANCOUVER, B.C., CANADA, V6T 122 DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF OTAGO

DUNEDIN, NEW ZEALAND E-mail: anstee@math. ubc.ca

ABSTRACT

For a general graph G and integer k , the paper proves the existence of a decomposition of G into k i G subgraphs whose degrees are close to of the corresponding degrees in G and each subgraph has approximately iJE(G)I edges. o 1996 John Wiley & Sons, Inc.

1. EXISTENCE OF FRACTIONS OF GRAPHS

In this paper we explore decompositions of graphs based on degrees using the idea of fractional subgraphs. Let G = ( V ( G ) , E(G)) be a graph (allowing multiple edges and loops). We use the terminology of Bondy and Murty 131. The first two results follow directly from a result of Kano [Thm 1,7]. We generalize a result of Petersen [8] which asserts that a graph G regular of degree 2k decomposes into k subgraphs each regular of degree 2. (Proofs of this result and others in this section are in Section 2. Euler tours are used.)

Theorem 1.1. Let G be a graph satisfying

d c ( v ) = O(mod 2 k ) , (1.1)

Then G has a subgraph (denoted i G satisfying

1 di,(v) = %dc;(v) for all 'u in V

Thus G can be edge decomposed into k such i G subgraphs. a Generalizations to arbitrary G have the problem that degrees congruent to k modulo 2k are

troublesome. For example a graph regular of degree k need not have a i G subgraph satisfying ( I .2). A little flexibility is required. The following result also follows from Kano [7] .

*Support provided by NSERC.

Journal of Graph Theory Vol. 23, No. 4, 377-384 (1 996) 0 1996 John Wiley & Sons, Inc. CCC 0364-9024/96/040377-08

378 JOURNAL OF GRAPH THEORY

Theorem 1.2. For an arbitrary graph G and integer k , G has f G subgraph satisfying (1.3), (1.4), (1.5). Moreover we can alter (1.4) by replacing +l by -1 for an arbitrary selection of vertices whose degrees are congruent to k modulo 2k.

1 k d:G(u) = -&(PI) if &(v) = O(mod 2 k ) , (1.3)

I

This is reminiscent of a number of results where the degree constraints get relaxed to obtain a general result. For example Tutte [9] showed that a graph regular of degree k has a subgraph whose degrees are T or T + 1 for any T < k. The case k = 2 is a bit kinder. A subgraph of G can be considered half of G, denoted i G , if

and for good measure

( 1.7)

We show that i G exists under the mildest possible conditions when (1.6) does not create obvious problems. This result appears to be well known (folklore?).

Theorem 1.3. Given a graph G, a subgraph i G exists satisfying (1.6) (1.7) if and only if no component of G has all vertices of even degree and an odd number of vertices whose degrees are

I

Note that the complement of aG in G is also half a graph providing a decomposition of G into 2 i G subgraphs. Might this be useful algorithmically as part of a divide and conquer strategy?

Our main new results generalize (1.7) to general k. In Section 3 we show the existence of a i G subgraph satisfying a version of (I .7). In Section 4 we obtain a decomposition of G into k i G subgraphs satisfying variants of (1.3)-( 1.3, ( I .7). Be warned that the notation i G is a shorthand that takes on several meanings in the paper. In each theorem the precise degree constraints are stated. What notion of i G to be used depends on what you wish to accomplish. Our approach uses fractional (g, f)-factors. We can encode a subgraph by an integral vector z = (2, : e E E(G)) where if A, is the multiplicity of e in G then for all e in E ( G )

( i G ) 1 = { I E ( G ) 1 / 2 if IE(G)I is even, IE(G)1/2 f 1 / 2 if IE(G)I is odd.

congruent to 2 modulo 4.

Given integral vectors g = (g (u ) : u E V ( G ) ) , f = ( f ( u ) : v E V ( G ) ) , a ( g , f)-factor is a subgraph 5 of G satisfying for all u E V ( G )

g(u> 5 &(u) 5 f(.). (1.9)

DIVIDING A GRAPH BY DEGREES 379

The existence of i G can be determined by graph factor algorithms. The algorithm of Bo Zhou [ 101 directly tackles conditions such as (1.7) for a (g, f)-factor. If we allow z, to be fractional and extend the definition of d, (w), we have what is called afractionuZ(g, f)-facror- The existence of a fractional (9, f)-factor is a network flow problem but in our problems can be determined directly, e.g., in Theorem 1 . 1 we might take z, = G. In certain cases, fractional (9, f)-factors yield (9, f)-factors and we exploit this idea in the proofs. We define the deficiency with respect to g, f of a fractional subgraph z as

A, to get a fractional subgraph

S:(f,g) = c M u ) - & ( ~ ) , O , d z ( v ) - f (v ) ) , (1.10) U E V

and S*(g, f ) is the minimum of (1.10) over all 5.

Proposition 1.4 [2]. There is a minimum deficiency fractional subgraph z of G with the property that z(e) is integral with the exception of the edges of a set of vertex disjoint odd cycles upon which 2 . z(e) is integral and each vertex u on the odd cycles has g ( u ) = f(u) = d,(u) .

Thus S* (9, f ) is an integer at least 0 and so if Si(f, g) < 1 we know a fractional (g, f)-factor exists.

2. EULER DECOMPOSITIONS ARE USEFUL

Two main tools are used, The first is a standard part of the study of Euler tours [4].

Theorem 2.1. Let G be a connected graph with 2k vertices of odd degree. If k = 0 then G has an Euler tour: a closed trail using all the edges. If k > 0 then the edges of G can be decomposed

I

The other result concerns fractional (9, f)-factors. Let G,=f denote the subgraph of G induced

into k trails joining all the odd degree vertices in pairs.

by those vertices w where g(w) = f ( w ) .

Theorem 2.2 [l]. Let G, g, f have the property that either (i) G,=f is bipartite or (ii) every pair of vertex disjoint cycles is joined by an edge, g(w) = f(v) for all w E V and EVE” f(w) is

I even. Then G has a (9, f)-factor if and only if G has a fractional (9, f)-factor.

The condition (ii) is mostly due to Fulkerson, Hoffman, and McAndrew [ 5 ] . Alternate Proof of Theorem 1.1.

each component and so by orienting each tour we get an orientation D of G with Since G has all degrees even then G has an Euler tour on

indegD(w) = outdeg,(v) = $ G ( w ) , (2.1)

$ d ~ ( w ) = O(mod k). (2.2)

We form a bipartite graph B = (V(B) , E ( B ) ) where V ( B ) = {w’ : 2) E V ( G ) } U {w” : w E V ( G ) } and we have X edges (u’, w”) E E ( B ) exactly when D has X directed edges (u, v). Thus the edges of B directly correspond to edges of G. We have

d ~ ( w ’ ) = outdegD(w),

d ~ ( d ’ ) = indegD(v). (2.3)


We replace each edge of B by a k fractional edge and so obtain a fractional subgraph whose degrees are k of those in B . By Theorem 2.2, B has an integral subgraph H whose degrees are those of B. Using (2.1) (2.2) (2.3) and regarding the edges of H as edges of G, we have obtained

I

Although G may have vertices of odd degree, we may obtain the Euler tour/Euler trail decomposition of each component of G using Theorem 2.1. Orient each Euler tourEuler trail into a directed tourkrail to obtain a directed graph D satisfying (2.1) for vertices of even degree and for vertices v of odd degree

indegI,(v) = d ~ . ( v ) / 2 f 1 / 2 ,

the subgraph i G . Induction on k yields the decomposition of G into k i G subgraphs.

Alternate Proof of Theorem 1.2.

outdegD(v) = d c ( v ) / 2 1 / 2 . (2.4)

Then we form the bipartite graph B = (V( B ) , E( B ) ) where V ( B ) = { v’ : ‘u E V ( G ) } U { v” : v E V ( G ) } and we have X edges (u’, v”) E E ( B ) exactly when D has X directed edges (u, u). Let

U = { ,u E V ( G ) : &(w) $ O(rnod 2 k ) } . (2.5)

Then for each vertex u E U we identify u’ and u” back into a single vertex u to obtain a new graph B. For w E V - U , we set

1 2k

gfv’) = f(v’) = g(,u”) = f(v”) = -dG.(v), (2.6)

and for u E U

(2.7)

Now we can obtain a fractional (g, f)-factor of B by replacing each edge by a fractional edge. Since B,q=f is bipartite we can use Theorem 2.2 to get a (9, f)-factor of B which yields, by the natural correspondence, a $G subgraph satisfying (1.3) (1.4) (1 3. There is some choice in the case of vertices u of degree congruent to k modulo 2k where we could take

for some arbitrary preselected set of such vertices. I

We relax the constraints for $G in Sections 3 , 4 to get results involving IE( $G)I.

Proof of Theorem 1.3. Assume G is connected with an even number of vertices of degrees congruent to 2 modulo 4. Use the oriented Euler tourEuler trails to obtain an orientation D of G satisfying (2.1) for vertices of even degree and (2.4) for vertices of odd degree. If there are no vertices of odd degree, then the Euler tour has an even number of edges and so we can take alternate edges of G along the Euler tour of G to obtain the desired :G subgraph. If there are vertices of odd degree, then take alternate edges of G along the Euler trails of G, but we do so as to preserve (1.7), namely if there are p trails with an odd number of edges then start with taking the first edge and start 151 with taking the second edge. This yields the desired subgraph $G.

The converse is elementary. I


In cases where i G does not exist, it is possible to use the above elementary proof to find a subgraph which satisfies (1.7) and also (1.6) with the exception of one vertex in each difficult component and then the degree need only be off by 1.

3. DIVIDING THE EDGES EVENLY

Looking at our fractional i G subgraph of Theorem 1.2 one might hope to find a $G subgraph satisfying

(3.1)

Theorems I . 1 and 1.3 give cases where it works. But what if a $G subgraph does not exist? Or what about general G? We cannot hope to do very well in general since if G is the triangle with each edge of multiplicity X then all the degrees are 2X yet a (&)G subgraph satisfying (1.3) (1.4) (1 S ) is a matching and so has at most one edge which is only & of the total edges. If we relax the degree constraints to intervals of size 3 we get a result.

Theorem 3.1. For an arbitrary graph G and integer k, a i G subgraph exists degrees satisfying (3.2)-(3.5) and number of edges satisfying (3.6). Also a i G subgraph exists degrees satisfying (3.2)-(3.5) and number of edges satisfying (3.7).

If d ~ ( v ) = 2kl + 1 then dt,(v) {21,21 + 1) . (3.3)

If d c ( v ) = 2k1 + t then dg,(zr) E {21,21, +1,21 f 2)for 2 5 t 5 2k - 2. (3.4)

If dc (w) = 2 k ( l + 1) - 1 then d l G ( w ) E (21 + 1,21+ 2). (3.5)

(3.6)

(3.7)

Proof. From G form the bipartite graph B = ( V ( B ) , E ( B ) ) as in the proof of Theorem 1.2. We extend B to a new graph B’ = ( V ( B ) U { s } , E(B’)) with E(B’) consisting of E ( B ) and

{(w, s ) : ds(w) f Ofmod k) >. (3.8)

For w E V ( B ) we set

(3.9)


This contrasts with (2.7). We form a fractional subgraph x’ by taking x’(e) = for all e E E(B) and by taking x’(w, s ) = [ i d n ( w ) l - kdg(w).

We use the edge (w, s) to take up the slack in our fractional solution so that (3.9) is fea- sible. Effectively we are determining a subgraph of B whose degree at u E V ( B ) is in { l i d ~ ( u ) J , [ i d ~ ( u ) 1 } . Now if &(v) = 21 then by (2.1) we are determining a subgraph of G whose degree at u is in { 2 1 ; ] , L i J + [:I, 2 [ i ] } . If ~ G ( ’ u ) = 21 + 1 then by (2.4) the degree at visin{[$] +I?], L i J + [?I, [i]+[F]}.Thesecorrespondto(3.4)withsomereductions occumng if lcll or k l ( l + 1) yielding (3.3) or (3.5). Let

Now if p is an integer with

(3.10)

(3.1 1)

being even then let g(s) = f ( s ) = p . We find that x’ is a fractional (g, f)-factor. Thus by Theorem 2.2 we find that B’ has a (9, f)-factor which yields the desired 5G subgraph. We note that if p = 0, then (3.1 1) is even since B is bipartite.

If p is not as above, then there is an integer q 2 0 so that q < p < q + 2 and yet

q < p < q + 2 and q + f ( u ) is even. v € V ( B )

(3.12)

There are now four cases to complete our construction. Case I . p > q + 1 and (3.6) We set g(s) = f ( s ) = q + 2 and compute that def,! (9, f ) < 1. By Prop. 1.4 we conclude

there is a fractional (9, f)-factor and then use Theorem 2.2 (ii) to get the desired (g, f)-factor noting that any odd cycle of B’ uses vertex s.

p < q + 1 and (3.7)

p 5 q + 1 and (3.6)

Case 2. We set g(s) = f ( s ) = q and compute that def,t (9, f ) < 1 and then proceed as in Case 1. Case 3. We extend B’ to a new graph B” = (V(B’) U { t} ,E(B’) U { ( s , t ) , ( t i t ) } . We set g(s) =

f ( s ) = q + 2 and g ( t ) = f ( t ) = 2 with a new fractional subgraph 5‘’ the same as x’ except x”(s, t ) = 1 and x”(t, t ) = f . We compute that def,jj(g, f ) < 1. By Prop. 1.4 we conclude there is a fractional (9, f)-factor and then use Theorem 2.2 (ii) to get the desired (9, f)-factor noting that the loop ( t , t ) is the only new odd cycle and it is joined to s which is on every odd cycle of B’. Now g(t) = f ( t ) = 2 forces the loop ( t , t ) to be used and the edge (s, t ) to be never used in every (9 , f)-factor and so we get the desired subgraph of B.

Case 4. WeextendB‘toanewgraphB’‘asinCase3. Wesetg(s) = f ( s ) = q+landg(t) = f ( t ) = 1

with a new fractional subgraph 2’’ the same as x’ except x”(s, t ) = 0 and x”(t, t ) = +. We compute that def,!! (9, f ) < 1 and then proceed as in Case 3. Now g ( t ) = f ( t ) = 1 forces the edge (s, t ) to be used and the loop ( t , t ) to be never used in every (9, f)-factor and so we get the

p 2 q + 1 and (3.7)

desired subgraph of B. I


4. DECOMPOSITIONS

It is reasonable to hope that there is a decomposition of G into k i G subgraphs, i G being suitably defined where we require the number of edges in each subgraph to be roughly IE( G) 1 . Careful applications of Theorem 3.1 yield the following.

Theorem 4.1. Given an arbitrary graph G and integer k, there is a decomposition of G into k i G subgraphs each satisfying (3.1)-(3.5).

We must carefully use Theorem 3.1 to get the decomposition into k kG subgraphs. Consider the decomposition of G into k i G subgraphs satisfying (3.2)-(3.5). We use induction on k. Choose a $G subgraph F1 using Theorem 3.1 leaving unspecified the choice (3.6) or (3.7) until we consider (3.1). Let G1 = G - F1. By induction G1 decomposes into k - 1 &GI subgraphs. F 2 , F3, . . . , F k . We verify that each &GI subgraph is a $G subgraph. If d ~ ( v ) = 2kl then d ( l / k ) ~ ( v ) = 21 by (3.2), dc , (v) = 2(k - 1)l and so d [ l , ( k - l ) ~ ~ l (v) = 21 by (3.2). If &(v) = 2kl+ 1 then d ( l / k p ( ~ ) E {21,21+ I} by (3.3), dcl (v) E {2(k - 1)1,2(k - 1)1+ 1) and so dp/(&1)]G1 (v) E {21,21+ l} by (3.2) (3.3). If &(v) = 2kl+ t for 2 5 t 5 2k - 2 then

and so d[1/(&1)]G1 (v) E {21,21 + 1,21 + 2) by (3.2) (if t = 2 or 2k - 2), by (3.3) (3.5) (if t = 2,3,2k - 3 or 2k - 2), and by (3.4). We need 2 5 t 5 25 - 2 to avoid getting other choices. Finally, if dc(v ) = 2k(l + 1) - 1 then d( l , ,+)~(v) E (21 + 1,21 + 2) by (3.3, dc,(v) E {2(k - 1)(1 + 1) - 1,2(k - 1)(1 + 1)) and so d [ l / ( ~ - l ) ~ ~ l ( v ) E (21 + 1,21 + 2) by (3.2) (3.5). Thus our &GI subgraphs are all i G subgraphs proving the existence of the decomposition.

A similar easier analysis yields (3.1). Recall that we have a choice between LIE(G)l/kj and [lE(G)l/kl in Theorem 3.1. Let FI, F 2 , . . . , F k denote the k kG subgraphs that get chosen as in the above argument and let

Prooj

d ( l / k ) ~ ( v ) E {21,21+1, 21+2} by(3.4),dc1 (u) E {2(k-l)l+t-2,2(k-l)l+t-l, 2(k-l)l+t)

Gi = G - Fi - F 2 - . . . - Fi. (4.1)

Assume JE(G)I = kp+lwithO 5 1 < ksothat [IE(G)l/k] = pand [IE(G)I/kl = p + l (unless 1 = 0). Choosethefirst1 F,'swithIE(F,)I =p+landtheremainingk-1 F,'swithIEF,)I = p . This makes sense since, for example, for i 5 1 - 1 we have IE(G,)I = kp + 1 - i(p + 1) =

I The following is a sample application. Let a [c, d]-graph be a graph whose degrees belong to

(k - i)p + (1 - i) and so l[E(G,)l/(k - i ) J = p and [IE(G,)[/(k - i ) ] = p + 1.

{ c , c + l , . . . , d } .

Corollary 4.2. (Kano [61). Let G be a [2an, 2bn]-graph. Then G can be decomposed into n [2a, 2b]-graphs.

Use Theorem 4.1. A i G subgraph satisfying (3.2)-(3.5) is a [2a, 2b]-graph. We Proof. ignore (3.1). I

ACKNOWLEDGMENTS

The author gratefully thanks the referees, particularly the referee who pointed out substantial improvements to the original versions of Theorems 3.1 and 4.1. The referee suggested a clever construction in order to follow the argument of Kano [7] but we prefer to follow the fractional to integral strategy.


References

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Dividing a graph by degrees

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