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On Disjoint Paths in Graphs
Annals of Discrete Mathematics 4 1 (1989) 333-340 0 Elsevier Science Publishers B.V. (North-Holland)
On Disjoint Paths in Graphs W. Mader
Institute of Mathematics University of Hanover Hanover, FRG
Dedicated to the memory of G. A . Dirac We survey some results on the maximum number of edge-disjoint paths in a graph, where a pair of upper bounds is assigned to each vertex, one for the number of paths ending in it, the other for the number of paths passing through it. As an application, we prove a conjecture due to M. Hager.
All graphs and multigraphs considered in this paper are supposed to be finite. Multigraphs may contain multiple edges, but no loops, whereas graphs have only simple edges. We consider a multigraph G in conjunction with two functions e and p from the vertex set V ( G ) to the non-negative integers N , and we denote such a ‘valued’ multigraph by Gg. In [l], T. Gallai posed the problem to determine the maximum number p(Gg) of edgedisjoint paths in a valued multigraph G: such that every vertex z is an endpoint of at most e(z) paths and an intermediate point of a t most p ( z ) paths. In this problem, and hence in this paper, a path always has length a t least 1 and different endpoints. The analogous problem, which arises by admitting also ‘closed paths’, was solved by T. Gallai in [l] (cf. also [7]). In [6],we determined p(G:) as the minimum of certain values of the partitions of V ( G ) , and we specialized this result by various choices of the functions e and p . We shall survey these results here and apply them to prove a conjecture of M. Hager [2]. Let us first fix some notation. The edge set of the graph G is denoted by E(G) and its set of components by L ( G ) . A C E L(G) is called trivial, if (CI = 1. The edge joining the vertices z and y in a graph is denoted by [z,y]. Let d ( z ; G ) be the degree of the vertex z in G. For disjoint subsets C1, , ,Ck of V ( G ) , K(CI,. . ,Ck;G) denotes the number of edges , Ck in G. For X & V ( G ) , let joining any two different classes C1, B ( X ; G )be the set of those vertices of X ,which are joined by an edge to
..
.
.. .
W. Mader
334
a t least one vertex of V ( G )- X , and let G ( X ) be the subgraph spanned by X in G . For x E V ( G ) , N ( x ; G ) denotes the set of neighbours of 2 in G. An x , y - p a t h in G is a path with (different) endpoints x and y, and for X , Y V ( G ) ,an X,Y-path is an x,y-path with x E X and y E Y . The domain of a function f is denoted by Dj. Let G be a graph and A V ( G ) . If we choose the functions e , p such that e ( x ) 2 d (x;G )and p(x) = 0 for all x E A and e ( x ) = 0 and p ( x ) = 1 for all x E V ( G )- A, then p(GZ) becomes the maximum number p ( A ;G) of openly disjoint A-paths in G, that means paths with both endpoints, but no intermediate point in A, which are disjoint on V ( G )- A . The determination of p ( A ; G )was the key for the solution of Gallai’s general problem. Obviously, we may confine ourselves to an independent A E V ( G ) , that means a set of vertices A such that E ( G ( A ) )= 8.
c
Theorem 1 [5]. For independent A
c V(G),
. ..
where the minimum is taken over all partitions (Co, C1, ,C,) of V (G)- A such that G - [CO U U E l E(G(Ci))]does not contain an A-path. The first proof of Theorem 1 was given in [5] by applying Menger’s Graph Theorem. (On the other hand, Theorem 1 easily implies Menger’s theorem.) In [3], L. Loviisz deduced a somewhat generaliaed version of Theorem 1 from his polymatroid matching theorem. He considered disjoint vertex sets A1,. . . ,Ak C V ( G ) ,and he determined the maximum number P(A1,... ,Ak; G ) of disjoint paths P in G, with each path P being an A;,Ajpath for any i # j. It is no problem to get one of the numbers p ( A ; G ) and P(A1,. ,Ak; G ) from the other. Given A = {al,. . . ,ah} E V ( G ) ,we split every vertex ai E A into an independent set Ai of d(ai;G) vertices of degree 1 to obtain a graph, which we denote by Then p ( A ; G ) = P(A1,. ,Ak; Vice versa, given disjoint vertex sets A l , . , Ak in G , define F by adding k new vertices { a l , ,a k } =: A to G and joining ai to all the vertices of Ai for i = 1, ... , k . Then P(A1,. ,Ak;G ) = p ( A ; It is also easy to get a formula for P(A1,. ,Ak; G ) , if the sets A1, . . . , AI, are not necessarily disjoint. We will derive such a formula only for k = 2.
..
..
z.
m.
Theorem 2. For A1,Az
.. . ..
c V(G),
..
..
q.
On Disjoint Paths in Graphs
335
where the minimum is taken over all partitions (Co,C1,. .. ,C,)of V ( G ) such that ICn(A1 uA2)l 2 2, but C n A 1 = 0 or C n A 2 = 0 holds for every non-trivial C E L(G - [COU UyZl E(G(Ci))]). Proof. Let A := {u1,a2} U { a , : z E A1 n A * } be a set of 2 elements such that A n V ( G )= 8 and define the graph G by and
+ IAl n A21
V ( c ):= V ( G )U A
E ( c ) := E(G)U { [ a l , x ] : x E A1 - A2 } U{[aa,~]:x€Az-A1} U{[a,,x]:xEA1nA2).
Then p(Al,A2;G) = p ( A ; g , and by Theorem 1 there is a partition (CO, C1,. .. ,Cn) of V ( G ) such that G - [COU U z 1 E(G(Ci))]does not contain an A-path and
holds. Choose such a partition (Co,C1,. possible. 0 bviously,
. . ,Cn)so that
n is as small as
B(Ci;G - Co) = B(C;; G - Co) U (Ci n (A1 U A2)). Let C be a non-trivial component of G - [COU U b l E(G(C;))].We may assume C n Ci = 0 for i = 1, . . . , k, and C n C; # 0 for i = Ic 1, . . . , n. As C is non-trivial, we have k I n - 2. Of course,
+
ci := c1, . . ., ci := c k , ci+1:= c k + l u is a partition of V ( G )- COsuch that - [COU U!:
* * *
u cn
E(G(C,'))]does not
contain an A-path. Let us suppose IC n (A1 U A2)l I 1. Then
W. Mader
336 and hence
n
C
I B ( C i + l ; E - C O )5~ l B ( C ; ; E - Co)I - ( n - k i=k+l
- 1).
But this contradicts the choice of n. Hence IC fl (A1 U A2)1 2 2, and C n A1 = 0 or C n A2 = 0, as G - [COU UyZl E(G(Ci))] does not contain an Al, Az-path. As on the other hand n
F(Ai,Az;G) I lcol t C [ $ l B ( C i ; G i=l
.
- Co) U (Ci n (A1 U A2))Il
for every (Co,C1,. , ,C,) satisfying the condition of Theorem 2, the proof 0 of Theorem 2 is complete.
.
If A1 = A2 in Theorem 2, the partitions (CO, C1,. . ,Cn) considered there have the property that all components of G - [Co U lJy.l E(G(Ci))] are trivial; hence, for i = 1, , . , n , Ci consists of components of G - Co. So Theorem 2 implies immediately the generalization of Tutte's famous l-factor theorem (i.e., the case A1 = A2 = V(G)) given by Gallai in [l]:
.
c
For independent A V(G), we consider the sets T C V(G) - A with the property that G - T does not contain an A-path; we call such a vertex set T A-separating and define .r(A;G) := min{ IT/ : T A-separating}. Of course, we have always p(A; G) 5 T(A; G), and Menger's theorem is the equality p(A; G) = T ( A ;G) for /A[ = 2. But obvious examples show that equality does not hold in general. For instance, let G arise from a complete graph K 2 n + l by adding an independent set A of 2n 1 vertices and joining these vertices by 2n 1 independent edges to K 2 n + l . Then p(A; G) = n = '.r(a;G). It was conjectured by T. Gallai in [l]that always p(A;G) 2 TTT(A;G). This conjecture is easily derived from Theorem 1.
+
+
Corollary 1 [6]. For every independent vertea: set A in a graph G,
p(A; G ) 2 ~ T ( AG). ;
If we define
c
T(A1,A2; G) := min{ IT1 : T V(G) such that G - T does not contain an A,,A,-path},
we get a similar corollary from Theorem 2.
On Disjoint Paths in Graphs
337
Corollary 2 . For d l vertex sets A1, A2 in F(A,,A2;G)
Q
graph G,
1 3%41,A2;G).
Proof. By Theorem 2, there is a partition (CO, C1,. . . ,Cn) of V(G) such that G - [CO U UZ1 E(G(Ci))] does not contain an Al,Az-path and
holds, where Bj := B(Cj;G - Co) U (Ci n (A1 U A2)) for i = 1, . . ., n. P ut B: := Bi,if lBjl is even, and Bi := Bj - {b;} for any bj E Bi, if IB;(is odd. Then every A1, A2-path in G contains a vertex of T := COU UZ1 Bi, and so we get n
'f(A1,A2;G) 5 IT1 5 2lcol
+ CIBII = 2@41,A2;G).
0
i=l
We shall point out now that Corollary 2 immediately implies Conjecture 1 of the paper [2]. First we need a concept introduced by L. Montejano and V. Neumann-Lara in [8]. For vertices 5 # y in a graph G, they defined
and
pzn(z7 y; G) := max{ IS1 : S system of openly disjoint x , y-paths of length a t least n } TZn(5,Y;G) := min{ IT1 : T V(G) - {x,Y} such that G - T has no z,y-path of length a t least n } ,
and they proved in [8] for all n 2 2,
For n = 3, M. Hager [2] improved this to
and he conjectured p23(5,?/;G)
1 +Tz3(5,Y;G).
This conjecture is immediately implied by Corollary 2 (and vice versa, too), because P23(5,Y; G) = P("; G),N(y; G); G - (5, Y})
W. Mader
338 and
713(5,9;G)= T ( N x ; G ) , N ( y ; G ) ; G- b , Y } ) for non-adjacent vertices x # y. Let us now return to Gallai's general problem, which was solved in [6] by applying Theorem 1 to a suitably modified graph. Let GZ be a valued multigraph, where e and p are functions from V ( G ) to N . As usual, we e(x) for X V ( G ) . The set of all (ordered) write e ( X ) instead of EXEX partitions (C,,,, Cm+l,. ,Cm+k) of V ( G )for any integers m 5 0 5 m k is denoted by P(G). Let P = (C,,,, ,Cn) E P(G). We call a function f : Df-+ { m ,.. .,-1) P-admissible, if
..
+
.. .
n
n
i=l
i= 1
U~ ( c -i u:=~ ; ~ Cj) E ~f E Uci
and f ( x ) = f ( y ) for all adjacent vertices x and y, which belong to different ones of the classes C1, . . . ,Cne For i > 0 and P-admissible f,
+ + + ~ ( c iUjj =1m C j ; G ) - C
vfp(Ci):= e(C;) e(Ci n Df) p(Ci n Dj)
~ ( 5~ f, ( x )G). ;
XECinDf
Furthermore,
The following theorem solves the above mentioned problem of T. Gallai.
Theorem 3 [6]. For every multigraph G valued by functions e , p from V ( G ) to N ,
If we choose the function p such that p(x) = 0 for all x E V ( G ) ,then
p(Gg) becomes the maximum number of edges in a (spanning) subgraph H
of G satisfying the condition d ( s ; H ) 5 e(x) for all x E V ( G ) . Hence, Theorem 3 yields Tutte's criterion for the existence of an e-factor [9](cf. Theorem 4 below), as the existence of an e-factor is equivalent to p(G$')2 i e ( V ( G ) ) .If we choose a set of two vertices A = { a l , a 2 } and the functions e and p as in the paragraph before Theorem 1, then p(GZ) = p( { a l , up};G).
Disjoint Paths in Graphs
339
So, Theorem 3 is a common generalization of Menger’s theorem and of Tutte’s theorem. Let us turn now to an analogue of Corollary 1. We call a path A C G (with different endpoints) admissible in GZ, if e ( z ) > 0 for both the endpoints z of A , and p(x) > 0 for all the intermediate vertices z of A. A triple (T,S, El) with T U S C V(G) and E’ E E(G) is called GZ-separating, if every admissible path in Gf has an endpoint in T , an intermediate vertex in S, or an edge in El. (For instance, if we take p as in the first sentence of the preceding paragraph, then (T,0,E’) is Gf-separating, iff (G - E’) - T does not contain adjacent vertices z and y with e ( z ) > 0 and e ( y ) > 0. If, in addition, e ( z ) = 1 for all z E V(G), then (T,0,0)is Gz-separating, iff T is a vertex cover, that means E(G - T ) = 0.) The inequality p(GZ) 5 r(GZ) := min{ e ( T )
+ p(S) + IE‘I : (T,S, El) Gf-separating}
is again obvious, but equality does not hold in general. But Theorem 3 gives the best lower bound for p(Gf).
Corollary 3 [6]. p(G{) 2 ~T(GZ). Specializing the functions e and p in different ways, one gets various interesting results (cf. [6]), but we will confine ourselves to one further case. Let A be a subset of V(G), and the functions e , p may satisfy the condition e ( z ) = 0 for all x E V(G) - A, p ( z ) = 0 for all x E A, and p ( z ) large enough, say p(z) 2 IE(G)I, for all x E V(G) - A. Then X(A; G+) := p(GE) depends only on the restriction elA of e to A and is the maximum number of edge-disjoint A-paths such that every x E A is contained in not more than e ( z ) of these paths. The following theorem is easily derived from Theorem 3.
Theorem 4 [el. Given a rnultigraph G, a subset A e: A --+ N , then X(A;G,) = min
V(G), and a function
/
(e(C0) +tc(Cl,...,Ck;G)
(co,ci,...,Ck)E’P(A;G)
where P(A;G):=
{ (CO,Cl,...,Ck):
Co,(71,. . .,Ck disjoint subsets of V(G) (k E N arbitrary) with Co A and ICi n A1 = 1 for all i = 1, .. . , k}.
W. Mader
3 40
Taking A = V(G) in Theorem 4, we have lCll = ... = Ickl = 1, and we get immediately Tutte’s general e-factor theorem. If we choose the function e so that e(x) is large enough for all x E A, then X(A;G,) is the maximum number of edge-disjoint A-paths. Then the minimum in Theorem 4 is attained by a (Co,Cl,. . . ,Ck)E P(A; G) satisfying Co = 8 and C n A = 0 for all C E L(G-U;=,C;);hence k = 1Al. We get, therefore, the main result of [4]: The maximum number of edge-disjoint A-paths is equal to min
{ ~(ci,. . . ,CI,;G) t Cc,=qc-Utlc,)L+(C, U L c i ; G ) J : k = IAJ and C1,.. . , CI, disjoint subsets of V(G) satisfying IC; n A! = 1 for dl i = 1, ... , k
}.
References [11 T. Gallai, “Maximum-Minimum-Satze und verallgemeinerte Faktoren von Graphen,” Acta Math. Acad. Sci. Hungar. 12 (1961), 131-173.
[2] M. Hager, “A Mengerian Theorem for paths of length at least three,” J. Graph Theory 10 (1986), 533-540.
[3] L. Lovisz, “Matroid matching and some applications,” J . Combin.Theory (B) 28 (1980), 208-236. [4] W. Mader, “Uber die Maximalzahl kantendisjunkter A-Wege,” Arch. der Math. 30 (1978), 325-336. [5] W. Mader, “Uber die Maximalzahl kreuzungsfreier H-Wege,” Arch. der Math. 31 (1978), 387-402. [6] W. Mader, “Uber ein graphentheoretisches Problem von T. Gallai,” Arch. der Math. 33 (1979), 239-257.
[7] W. Mader, “Uber ein graphentheoretisches Ergebnis von T. Gallai,” Acta Math. Acad. Sci. Hungar. 35 (1980), 205-215. [8] L. Montejano and V. Neumann-Lara, “A variation of Menger’s theorem for long paths,” J. Combin. Theory (B) 36 (1984), 213-217.
[9]W. T. Tutte, “The factors of graphs,” Canad. J. Math, 4 (1952), 3 14-328.
On Disjoint Paths in Graphs W. Mader
Institute of Mathematics University of Hanover Hanover, FRG
Dedicated to the memory of G. A . Dirac We survey some results on the maximum number of edge-disjoint paths in a graph, where a pair of upper bounds is assigned to each vertex, one for the number of paths ending in it, the other for the number of paths passing through it. As an application, we prove a conjecture due to M. Hager.
All graphs and multigraphs considered in this paper are supposed to be finite. Multigraphs may contain multiple edges, but no loops, whereas graphs have only simple edges. We consider a multigraph G in conjunction with two functions e and p from the vertex set V ( G ) to the non-negative integers N , and we denote such a ‘valued’ multigraph by Gg. In [l], T. Gallai posed the problem to determine the maximum number p(Gg) of edgedisjoint paths in a valued multigraph G: such that every vertex z is an endpoint of at most e(z) paths and an intermediate point of a t most p ( z ) paths. In this problem, and hence in this paper, a path always has length a t least 1 and different endpoints. The analogous problem, which arises by admitting also ‘closed paths’, was solved by T. Gallai in [l] (cf. also [7]). In [6],we determined p(G:) as the minimum of certain values of the partitions of V ( G ) , and we specialized this result by various choices of the functions e and p . We shall survey these results here and apply them to prove a conjecture of M. Hager [2]. Let us first fix some notation. The edge set of the graph G is denoted by E(G) and its set of components by L ( G ) . A C E L(G) is called trivial, if (CI = 1. The edge joining the vertices z and y in a graph is denoted by [z,y]. Let d ( z ; G ) be the degree of the vertex z in G. For disjoint subsets C1, , ,Ck of V ( G ) , K(CI,. . ,Ck;G) denotes the number of edges , Ck in G. For X & V ( G ) , let joining any two different classes C1, B ( X ; G )be the set of those vertices of X ,which are joined by an edge to
..
.
.. .
W. Mader
334
a t least one vertex of V ( G )- X , and let G ( X ) be the subgraph spanned by X in G . For x E V ( G ) , N ( x ; G ) denotes the set of neighbours of 2 in G. An x , y - p a t h in G is a path with (different) endpoints x and y, and for X , Y V ( G ) ,an X,Y-path is an x,y-path with x E X and y E Y . The domain of a function f is denoted by Dj. Let G be a graph and A V ( G ) . If we choose the functions e , p such that e ( x ) 2 d (x;G )and p(x) = 0 for all x E A and e ( x ) = 0 and p ( x ) = 1 for all x E V ( G )- A, then p(GZ) becomes the maximum number p ( A ;G) of openly disjoint A-paths in G, that means paths with both endpoints, but no intermediate point in A, which are disjoint on V ( G )- A . The determination of p ( A ; G )was the key for the solution of Gallai’s general problem. Obviously, we may confine ourselves to an independent A E V ( G ) , that means a set of vertices A such that E ( G ( A ) )= 8.
c
Theorem 1 [5]. For independent A
c V(G),
. ..
where the minimum is taken over all partitions (Co, C1, ,C,) of V (G)- A such that G - [CO U U E l E(G(Ci))]does not contain an A-path. The first proof of Theorem 1 was given in [5] by applying Menger’s Graph Theorem. (On the other hand, Theorem 1 easily implies Menger’s theorem.) In [3], L. Loviisz deduced a somewhat generaliaed version of Theorem 1 from his polymatroid matching theorem. He considered disjoint vertex sets A1,. . . ,Ak C V ( G ) ,and he determined the maximum number P(A1,... ,Ak; G ) of disjoint paths P in G, with each path P being an A;,Ajpath for any i # j. It is no problem to get one of the numbers p ( A ; G ) and P(A1,. ,Ak; G ) from the other. Given A = {al,. . . ,ah} E V ( G ) ,we split every vertex ai E A into an independent set Ai of d(ai;G) vertices of degree 1 to obtain a graph, which we denote by Then p ( A ; G ) = P(A1,. ,Ak; Vice versa, given disjoint vertex sets A l , . , Ak in G , define F by adding k new vertices { a l , ,a k } =: A to G and joining ai to all the vertices of Ai for i = 1, ... , k . Then P(A1,. ,Ak;G ) = p ( A ; It is also easy to get a formula for P(A1,. ,Ak; G ) , if the sets A1, . . . , AI, are not necessarily disjoint. We will derive such a formula only for k = 2.
..
..
z.
m.
Theorem 2. For A1,Az
.. . ..
c V(G),
..
..
q.
On Disjoint Paths in Graphs
335
where the minimum is taken over all partitions (Co,C1,. .. ,C,)of V ( G ) such that ICn(A1 uA2)l 2 2, but C n A 1 = 0 or C n A 2 = 0 holds for every non-trivial C E L(G - [COU UyZl E(G(Ci))]). Proof. Let A := {u1,a2} U { a , : z E A1 n A * } be a set of 2 elements such that A n V ( G )= 8 and define the graph G by and
+ IAl n A21
V ( c ):= V ( G )U A
E ( c ) := E(G)U { [ a l , x ] : x E A1 - A2 } U{[aa,~]:x€Az-A1} U{[a,,x]:xEA1nA2).
Then p(Al,A2;G) = p ( A ; g , and by Theorem 1 there is a partition (CO, C1,. .. ,Cn) of V ( G ) such that G - [COU U z 1 E(G(Ci))]does not contain an A-path and
holds. Choose such a partition (Co,C1,. possible. 0 bviously,
. . ,Cn)so that
n is as small as
B(Ci;G - Co) = B(C;; G - Co) U (Ci n (A1 U A2)). Let C be a non-trivial component of G - [COU U b l E(G(C;))].We may assume C n Ci = 0 for i = 1, . . . , k, and C n C; # 0 for i = Ic 1, . . . , n. As C is non-trivial, we have k I n - 2. Of course,
+
ci := c1, . . ., ci := c k , ci+1:= c k + l u is a partition of V ( G )- COsuch that - [COU U!:
* * *
u cn
E(G(C,'))]does not
contain an A-path. Let us suppose IC n (A1 U A2)l I 1. Then
W. Mader
336 and hence
n
C
I B ( C i + l ; E - C O )5~ l B ( C ; ; E - Co)I - ( n - k i=k+l
- 1).
But this contradicts the choice of n. Hence IC fl (A1 U A2)1 2 2, and C n A1 = 0 or C n A2 = 0, as G - [COU UyZl E(G(Ci))] does not contain an Al, Az-path. As on the other hand n
F(Ai,Az;G) I lcol t C [ $ l B ( C i ; G i=l
.
- Co) U (Ci n (A1 U A2))Il
for every (Co,C1,. , ,C,) satisfying the condition of Theorem 2, the proof 0 of Theorem 2 is complete.
.
If A1 = A2 in Theorem 2, the partitions (CO, C1,. . ,Cn) considered there have the property that all components of G - [Co U lJy.l E(G(Ci))] are trivial; hence, for i = 1, , . , n , Ci consists of components of G - Co. So Theorem 2 implies immediately the generalization of Tutte's famous l-factor theorem (i.e., the case A1 = A2 = V(G)) given by Gallai in [l]:
.
c
For independent A V(G), we consider the sets T C V(G) - A with the property that G - T does not contain an A-path; we call such a vertex set T A-separating and define .r(A;G) := min{ IT/ : T A-separating}. Of course, we have always p(A; G) 5 T(A; G), and Menger's theorem is the equality p(A; G) = T ( A ;G) for /A[ = 2. But obvious examples show that equality does not hold in general. For instance, let G arise from a complete graph K 2 n + l by adding an independent set A of 2n 1 vertices and joining these vertices by 2n 1 independent edges to K 2 n + l . Then p(A; G) = n = '.r(a;G). It was conjectured by T. Gallai in [l]that always p(A;G) 2 TTT(A;G). This conjecture is easily derived from Theorem 1.
+
+
Corollary 1 [6]. For every independent vertea: set A in a graph G,
p(A; G ) 2 ~ T ( AG). ;
If we define
c
T(A1,A2; G) := min{ IT1 : T V(G) such that G - T does not contain an A,,A,-path},
we get a similar corollary from Theorem 2.
On Disjoint Paths in Graphs
337
Corollary 2 . For d l vertex sets A1, A2 in F(A,,A2;G)
Q
graph G,
1 3%41,A2;G).
Proof. By Theorem 2, there is a partition (CO, C1,. . . ,Cn) of V(G) such that G - [CO U UZ1 E(G(Ci))] does not contain an Al,Az-path and
holds, where Bj := B(Cj;G - Co) U (Ci n (A1 U A2)) for i = 1, . . ., n. P ut B: := Bi,if lBjl is even, and Bi := Bj - {b;} for any bj E Bi, if IB;(is odd. Then every A1, A2-path in G contains a vertex of T := COU UZ1 Bi, and so we get n
'f(A1,A2;G) 5 IT1 5 2lcol
+ CIBII = 2@41,A2;G).
0
i=l
We shall point out now that Corollary 2 immediately implies Conjecture 1 of the paper [2]. First we need a concept introduced by L. Montejano and V. Neumann-Lara in [8]. For vertices 5 # y in a graph G, they defined
and
pzn(z7 y; G) := max{ IS1 : S system of openly disjoint x , y-paths of length a t least n } TZn(5,Y;G) := min{ IT1 : T V(G) - {x,Y} such that G - T has no z,y-path of length a t least n } ,
and they proved in [8] for all n 2 2,
For n = 3, M. Hager [2] improved this to
and he conjectured p23(5,?/;G)
1 +Tz3(5,Y;G).
This conjecture is immediately implied by Corollary 2 (and vice versa, too), because P23(5,Y; G) = P("; G),N(y; G); G - (5, Y})
W. Mader
338 and
713(5,9;G)= T ( N x ; G ) , N ( y ; G ) ; G- b , Y } ) for non-adjacent vertices x # y. Let us now return to Gallai's general problem, which was solved in [6] by applying Theorem 1 to a suitably modified graph. Let GZ be a valued multigraph, where e and p are functions from V ( G ) to N . As usual, we e(x) for X V ( G ) . The set of all (ordered) write e ( X ) instead of EXEX partitions (C,,,, Cm+l,. ,Cm+k) of V ( G )for any integers m 5 0 5 m k is denoted by P(G). Let P = (C,,,, ,Cn) E P(G). We call a function f : Df-+ { m ,.. .,-1) P-admissible, if
..
+
.. .
n
n
i=l
i= 1
U~ ( c -i u:=~ ; ~ Cj) E ~f E Uci
and f ( x ) = f ( y ) for all adjacent vertices x and y, which belong to different ones of the classes C1, . . . ,Cne For i > 0 and P-admissible f,
+ + + ~ ( c iUjj =1m C j ; G ) - C
vfp(Ci):= e(C;) e(Ci n Df) p(Ci n Dj)
~ ( 5~ f, ( x )G). ;
XECinDf
Furthermore,
The following theorem solves the above mentioned problem of T. Gallai.
Theorem 3 [6]. For every multigraph G valued by functions e , p from V ( G ) to N ,
If we choose the function p such that p(x) = 0 for all x E V ( G ) ,then
p(Gg) becomes the maximum number of edges in a (spanning) subgraph H
of G satisfying the condition d ( s ; H ) 5 e(x) for all x E V ( G ) . Hence, Theorem 3 yields Tutte's criterion for the existence of an e-factor [9](cf. Theorem 4 below), as the existence of an e-factor is equivalent to p(G$')2 i e ( V ( G ) ) .If we choose a set of two vertices A = { a l , a 2 } and the functions e and p as in the paragraph before Theorem 1, then p(GZ) = p( { a l , up};G).
Disjoint Paths in Graphs
339
So, Theorem 3 is a common generalization of Menger’s theorem and of Tutte’s theorem. Let us turn now to an analogue of Corollary 1. We call a path A C G (with different endpoints) admissible in GZ, if e ( z ) > 0 for both the endpoints z of A , and p(x) > 0 for all the intermediate vertices z of A. A triple (T,S, El) with T U S C V(G) and E’ E E(G) is called GZ-separating, if every admissible path in Gf has an endpoint in T , an intermediate vertex in S, or an edge in El. (For instance, if we take p as in the first sentence of the preceding paragraph, then (T,0,E’) is Gf-separating, iff (G - E’) - T does not contain adjacent vertices z and y with e ( z ) > 0 and e ( y ) > 0. If, in addition, e ( z ) = 1 for all z E V(G), then (T,0,0)is Gz-separating, iff T is a vertex cover, that means E(G - T ) = 0.) The inequality p(GZ) 5 r(GZ) := min{ e ( T )
+ p(S) + IE‘I : (T,S, El) Gf-separating}
is again obvious, but equality does not hold in general. But Theorem 3 gives the best lower bound for p(Gf).
Corollary 3 [6]. p(G{) 2 ~T(GZ). Specializing the functions e and p in different ways, one gets various interesting results (cf. [6]), but we will confine ourselves to one further case. Let A be a subset of V(G), and the functions e , p may satisfy the condition e ( z ) = 0 for all x E V(G) - A, p ( z ) = 0 for all x E A, and p ( z ) large enough, say p(z) 2 IE(G)I, for all x E V(G) - A. Then X(A; G+) := p(GE) depends only on the restriction elA of e to A and is the maximum number of edge-disjoint A-paths such that every x E A is contained in not more than e ( z ) of these paths. The following theorem is easily derived from Theorem 3.
Theorem 4 [el. Given a rnultigraph G, a subset A e: A --+ N , then X(A;G,) = min
V(G), and a function
/
(e(C0) +tc(Cl,...,Ck;G)
(co,ci,...,Ck)E’P(A;G)
where P(A;G):=
{ (CO,Cl,...,Ck):
Co,(71,. . .,Ck disjoint subsets of V(G) (k E N arbitrary) with Co A and ICi n A1 = 1 for all i = 1, .. . , k}.
W. Mader
3 40
Taking A = V(G) in Theorem 4, we have lCll = ... = Ickl = 1, and we get immediately Tutte’s general e-factor theorem. If we choose the function e so that e(x) is large enough for all x E A, then X(A;G,) is the maximum number of edge-disjoint A-paths. Then the minimum in Theorem 4 is attained by a (Co,Cl,. . . ,Ck)E P(A; G) satisfying Co = 8 and C n A = 0 for all C E L(G-U;=,C;);hence k = 1Al. We get, therefore, the main result of [4]: The maximum number of edge-disjoint A-paths is equal to min
{ ~(ci,. . . ,CI,;G) t Cc,=qc-Utlc,)L+(C, U L c i ; G ) J : k = IAJ and C1,.. . , CI, disjoint subsets of V(G) satisfying IC; n A! = 1 for dl i = 1, ... , k
}.
References [11 T. Gallai, “Maximum-Minimum-Satze und verallgemeinerte Faktoren von Graphen,” Acta Math. Acad. Sci. Hungar. 12 (1961), 131-173.
[2] M. Hager, “A Mengerian Theorem for paths of length at least three,” J. Graph Theory 10 (1986), 533-540.
[3] L. Lovisz, “Matroid matching and some applications,” J . Combin.Theory (B) 28 (1980), 208-236. [4] W. Mader, “Uber die Maximalzahl kantendisjunkter A-Wege,” Arch. der Math. 30 (1978), 325-336. [5] W. Mader, “Uber die Maximalzahl kreuzungsfreier H-Wege,” Arch. der Math. 31 (1978), 387-402. [6] W. Mader, “Uber ein graphentheoretisches Problem von T. Gallai,” Arch. der Math. 33 (1979), 239-257.
[7] W. Mader, “Uber ein graphentheoretisches Ergebnis von T. Gallai,” Acta Math. Acad. Sci. Hungar. 35 (1980), 205-215. [8] L. Montejano and V. Neumann-Lara, “A variation of Menger’s theorem for long paths,” J. Combin. Theory (B) 36 (1984), 213-217.
[9]W. T. Tutte, “The factors of graphs,” Canad. J. Math, 4 (1952), 3 14-328.