Let G = (A, B; E) be a bipartite graph. Let e1, e2 be nonnegative integers, and f1, f2 nonnegative integer-valued functions on V(G) such that ei ≤|E|≤ e1 + e2 and fi(v)≤d(v)≤f1(v) + f2(v) for all v ε V(G) (i = 1, 2). Necessary and sufficient conditions are obtained for G to admit a decomposition in spanning subgraphs G1 = (A, B; E1) and G2 = (A, B; E2) such that |Ei|≤ei and dGi(v)≤fi(v) for all v ε V(G) (i = 1, 2). The result generalizes a known characterization of bipartite graphs with a k-factor. Its proof uses flow theory and is a refinement of the proof of an analogous result due to Folkman and Fulkerson. By applying corresponding flow algorithms, the described decomposition can be found in polynomial time if it exists. As an application, an assignment problem is solved

Broersma, H.J.

Faudree, R.J.

Heuvel, J. van den

Veldman, H.J.

English

Broersma, Haitze J.

van den Heuvel, J.P.M.

NARCIS 

Decomposition of bipartite graphs under degree constraints

University of Twente Research Information

Decomposition of Bipartite Graphs under Degree Constraints H. J. Broersma Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands R. J. Faudree Department of Mathematical Sciences, Memphis State University, Memphis, Tennessee 38152 J. van den Heuvel and H. J. Veldman Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands Let G = (A, 6;  E )  be a bipartite graph. Let el, e2 be nonnegative integers, and f l ,  f2 nonnegative integer-valued functions on V(G) such that ei I I € \  5 el + e2 and f f ( v )  5 d ( v )  5 f , (v )  + f2(v) for all v E V(G) (i = 1, 2). Necessary and sufficient conditions are obtained for G to admit a decomposition in spanning subgraphs GI = (A, 8 ;  El) and G2 = (A, 6;  E2) such that lEil I el and da,(v) 5 f i ( V )  for all v E V(G)  (i = 1,2). The result generalizes a known characterization of bipartite graphs with a k-factor. Its proof uses flow theory and is a refinement of the proof of an analogous result due to Folkman and Fulkerson. By applying corresponding flow algorithms, the described decomposition can be found in polynomial time if it exists. As an application, an assign- ment problem is solved. o 7993 by ~ o h n  wi/ey & Sons, Inc. 1. AN ASSIGNMENT PROBLEM The following problem was posed in van Doorn (61. In a certain year, each of k students at some school is required to spend probationary periods pl , . . . , p,n at m different companies from a collection {c, , . . . , c,} of n companies. In the period p i .  company cj can host at most rii students (i = I ,  . . . , m ; j  = 1, . . . , n) .  The question is whether and how the probation require- ments can be satisfied for all students. After introducing suitable notation and terminology in Section 2, we present in Section 3 a solution of this problem using the theory of k-factors in bipartite graphs. In Section 4, the results in Section 3 are gener- alized to an analog of a theorem due to Folkman and Fulkerson [2]. Using this generalization, another solu- tion of the problem, not using k-factors, is presented in Section 5 .  2. NOTATION AND TERMINOLOGY We use Bondy and Murty [ I ]  for notation and termi- nology not defined here. The graphs we consider may have multiple edges, but no loops. Let G be a bipartite graph with bipartition (A, B). We write G = (A, B ) ,  or G = (A, B; E )  if G has edge set E. The graph G is balanced if IAl = IBI. For X A and NETWORKS, Vol. 23 (1993) 159-164 0 1993 by John Wiley 8 Sons, Inc. CCC 0028-3045/93/030159-06 159 160 BROERSMA El AL. Y C B, qc(X, Y )  or q ( X ,  Y )  denotes the number of edges with one end in X and the other in Y. For y E Y, we write q ( X ,  y )  instead of 4(X ,  { y } ) .  If M is a matching in G ,  then M is S-saturating if every vertex of S is incident with an edge of M. For a nonnegative integer k ,  a k-factor of G is a k-regular spanning subgraph of G. More generally, for a nonnegative integer-valued function f o n  V(G), an f-factor is a spanning subgraph G' of G such that dc,(v) = f l u )  for all u E V(G). The function f is balanced if ZxEA x) = Z y E ~  f l y ) .  3. SOLUTION OF THE PROBLEM USING k-FACTORS Consider the following question: (1)  Given a bipartite graph G = (A,  B )  and a nonnega- tive integer k, do there exist k pairwise disjoint A- saturating matchings? As shown in van Doorn [6], the assignment problem of Section 1 can be translated into ( 1 )  by defining a bipar- tite graph G = (A, B) with A = { P I ,  . . . , p,,,} and B = { c ,  , . . . , c,,} in which the vertices p i  and cj are joined by rii edges ( i  = 1,  . . . , m; j - 1, . . . , n ) .  The probation requirements can be satisfied for all k stu- dents if and only if G admits k pairwise disjoint A- saturating matchings. A necessary and sufficient condition for (1) to have an affirmative answer can be derived from the follow- ing result onf-factors, attributed in Katerinis [4] to Ore and Ryser. Theorem 1. (Ore-Ryserf-factor theorem, see [41). Let G = (A, B )  be a bipartite graph and f a balanced nonnegative integer-valued function on V(G). Then, G has an f-factor if and only V, for all X C A ,  An immediate consequence of Theorem 1 is the fol- lowing: Corollary 2. Let G = (A, B )  be a balanced bipartite graph and k a nonnegative integer. Then, G has a k- factor if and only if, for all X C A ,  (3) Now the following consequence of Corollary 2 can be used to answer question (1): Corollary 3. Let G = (A ,  B )  be a bipartite graph, and k a nonnegative integer. Then, G has k pairwise disjoint A-saturating matchings if and only if, for all X C A ,  c m i n k  9 ( X ,  Y)) 2 klXI. (4) YEB Proof. We may assume IAl 5 IS]. Construct a bal- anced bipartite graph G* = (A*, B )  from G = (A, B )  by adding (B(  - (A1 new vertices, each of which is joined by k edges to each vertex of B. Then, (4) holds for all X C A if and only if, for all X* C A*, (Note that if X* contains a vertex of A* - A, then the left-hand side of ( 5 )  equals klBI.) We now show that G has k pairwise disjoint A-saturating matchings if and only if G* has a k-factor; hence, Corollary 3 follows from Corollary 2. Clearly, if G has k pairwise disjoint A-saturating matchings, then G* has k pairwise dis- joint perfect matchings and, hence, a k-factor. Con- versely, if G* has a k-factor H*, then G has a spanning subgraph H such that &(x) = k for all x E A and dH(y)  5 k for all y E B. By 11,  Theorem 6. I], H admits a k- edge-coloring. The color classes associated with such a coloring constitute k pairwise disjoint A-saturating matchings of G. Note that for k = 1, Corollary 3 reduces to Hall's Theorem [I, Theorem 5.21. Theorem 1 is proved via the max-flow min-cut theo- rem (see Ford and Fulkerson [3]). By using the associ- ated labeling method in 131, question ( 1 )  can be an- swered in polynomial time. Moreover, a collection of k pairwise disjoint A-saturating matchings in a bipartite graph G = (A, B )  can actually be constructed in poly- nomial time if it exists. 4. A GENERALIZATION OF THE ORE-RYSER FFACTOR THEOREM Our main result is the following: Theorem 4. Let G = ( A ,  B ;  E) be a bipartite graph with e = (El. Let fi , fi be nonnegative integer-valued functions on V(G) ,  and e l ,  e2 nonnegative integers sat- isfring e i 5  e 5 el + e2 ( i  = 1, 2 ) .  DECOMPOSITION OF BIPARTITE GRAPHS 161 Then, there exist spanning subgraphs GI = ( A ,  B ;  E l )  and G2 = ( A ,  B ;  E2) of G satisfying El U E2 = E ,  El fl EZ = 0, f u n d  only $,for all X A ,  Proof. Suppose (6) holds for spanning subgraphs GI = (A, B; E l )  and G2 = (A, B; E2) of G, and let X C A. Then, Also, On the other hand, It follows that (7) and (8) hold. Similarly, (9) and (10) hold. Conversely, assume that (7)-( 10) hold. We con- struct a network G’ from G by orienting all edges of G from A to B and adding two new vertices a and b together with the edges (6, a), (a, x) for each x E A, and (y, b) for each y E B. We define nonnegative inte- ger-valued functions I and c on E(G’) as follows: An integer-valued function f on the edge set of G’ is called a feasible flow if Kirchhoff s law is satisfied at each vertex of G’ and I(u, u) <f lu ,  u) 5 c(u, u )  for each edge (u,  u)  of G’. Iff is a feasible flow, then the span- ning subgraphs GI = (A,  B; E l )  and G2 = (A, B; ,521 of G with El = {(x, y )  E EIflx, y )  = I} and E2 = {(x, y )  E Elflx, y )  = O}, respectively, satisfy (6). (For example, I&( = e - IElI = e - C(x,y)EE,flx, y )  = e - CxEAfla,  x) = e - f i b ,  a )  5 e - I(b, a) = e2 and, for x E A ,  dcl(x) = f l u ,  x) 5 c(a,  x) = f i ( x ) . )  We will establish the exis- tence of a feasible flow. By [3, Theorem 3. I] ,  a feasible flow exists if and only if, for all Z C V(C’) ,  where .?? = V(G’) - Z. Let z then by (7), v(G’),  x = z n A ,  Y = z n B. I f a ,  6 E Z, UEZ If a E Z, b E 2, then by (8), 162 BROERSMA E l  AL. If a E 2, h E 2, then by (lo), fib) =flu) for all u E V(G),  hence, = l(u,  u ) .  NE.7 U E L  Finally, if a ,  b E Z ,  then by (91, hence, UEZ Having established (11)  for all Z V(G') ,  we are done. rn As we show next, Theorem I can be obtained from Theorem 4. Set e l  = e l  = e .  Then (7) amounts to (2), while (8) and (10) are trivially satisfied. The inequality (9), too, is trivially satisfied, since Hence under (21, G contains, by Theorem 4, a span- ning subgraph GI = (A, B; E l )  such that lEll = e l ,  dG,(x) = flx) for all x E A and dc,(y) I f l y )  for all y E B. Since &A d&) = C,EB d ~ , ( y )  and 2 , d l x )  = %EB fly),  in fact, dc,(y) = f ly)  for all y E B, implying that GI is anf-factor of C .  Thus, the sufficiency of (2) in Theorem 1 follows from the sufficiency of (7)-(10) in Theorem 4. Similarly, the necessity of (2) in Theorem 1 follows from the necessity of (7)-( 10) in Theorem 4. The proof of Theorem 4 is a refinement of the proof of the following analogous result. Theorem 5 (Folkman and Fulkerson [2]). Let G = ( A ,  B; E )  be u bipartite graph with e = /El. L e t f i . h  b e  nonnegative integer-uuluedfuncfions on V(C), and el , e2 nonnegative integers, sa t i sb ing  c = e l  + e 2 .  Then, there exist spanning subgraphs GI = ( A ,  B ;  El) and G2 = ( A ,  B; E2) of G sa t i sb ing  E~ u E? = E ,  E, n E* = 0, IE;J = ej ( i  = I ,  2 ) ,  DECOMPOSITION OF BIPARTITE GRAPHS 163 Note that Theorem 4 is of slightly wider applicability than is Theorem 5 because of the weaker restrictions on e l  and ez .  The conditionJ;(v) 5 d(v )  (v E V(G), i = 1, 2), occurring in Theorem 4 but not in Theorem 5, is no real restriction; it merely serves to give Theorem 4 a nicer shape. Theorem 5 has consequences analogous to Theo- rem 1, Corollary 2, and Corollary 3. The analog of Theorem 1 is in Lovasz and Plummer [5, Theorem 2.4.21. Here, we only mention the analog of Corollary 3. Corollary 6 .  Let G = (A, B )  be a bipartite graph and k a nonnegative integer. Then, G has k pairwise disjoint A-saturating matchings if and only if, for all X C A and Y C B, Problem (1) is “one-sided” in the sense that saturation requirements are made only with respect to A.  There- fore, the one-sided condition (4) in Corollary 3 may be considered more natural than the two-sided condition (12) in Corollary 6. Note in this connection that Hall’s Theorem is not immediate from Corollary 6. 5. SOLUTION OF THE PROBLEM WITHOUT THE USE OF &-FACTORS From Theorem 4 we reprove Corollary 3 without the use of k-factors. As the first proof of Corollary 3, this proof implies a (flow) algorithm that decides whether and how a collection of k pairwise disjoint A-saturating matchings in a bipartite graph G = (A, B) can be found. Alternative Proof of Corollury 3. Call a spanning subgraph H of G k-suitable if dH(x) = k for all x E A and d d y )  5 k for all y E B .  We establish two claims: (13) G contains a k-suitable spanning subgraph if and only if (4) holds for all X A .  Set Then, two spanning subgraphs G I  = (A, B ;  El) and G2 = (A, B ;  Ez) of G with El  U E2 = E and El n Ez = 0 satisfy (6) if and only if GI is k-suitable. Furthermore, (7)-(10) hold if and only if (4) holds. Thus, (13) follows from Theorem 4. (14) The edge set of a k-suitable spanning subgraph of G can be partitioned into the edge set of a ( k  - 1)- suitable spanning subgraph and an A-saturating match- ing. Let H be a k-suitable spanning subgraph of G and set hl(v) = 1 for all v E V(H),  i f v E A  r - ’  min{k - 1, d d v ) }  if v E B h 2 ( ~ )  =Let X C A. Clearly, since H is k-suitable, ( N d X ) (  I 1x1. Hence, Set B I  = { y  E B(9dX, y )  = k}  and B2 = B - B I  . Since H is k-suitable, lBll I (XI. Hence, fLv) = k for all v E V(G), By Theorem 4, (15) and (16) imply the existence of two spanning subgraphs H I  and H2 of H such that [:d(;-k i f v E A  h(V) = i f v E B  EW) u E ( H ~ )  = E M ,  E ( H ~ )  n E ( H ~ )  = 0, e l  = e2 = e .  dH,(v) 5 hi(v) (v E V(W, i = 1, 2), 164 BROERSMA ET AL. H2 is ( k  - I)-suitable and E ( H I )  is an A-saturating matching, proving (14). Corollary 3 follows from (13) and (14) via induction on k .  U We note that (14) could have been deduced immedi- ately from the fact that a k-suitable subgraph of G is k- edge-colorable. We chose to use Theorem 4 since it is applicable in more general contexts also. For example, our second proof of Corollary 3 is easily extended to a proof of the following result, where an A-saturating s- claw-factor of G = (A, B )  is defined as the edge set of a spanning subgraph H of G such that d d x )  = s for all x E A and d d y )  I I for all y E B. Theorem 7 .  Let G = (A ,  B)  be a bipartite graph and k ,  s nonnegative integers. Then, G has kpairwise disjoint A-saturating s-claw-factors i f  and only $,for all X c A,  REFERENCES J. A. Bondy and U. S .  R. Murty, Graph Theory with Applications. Macmillan, London, Elsevier, New York ( 1976). J. Folkman and D. R. Fulkerson, Edge colorings in bipartite graphs. Combinatorial Mathematics and Its Applications. (R. C .  Bose and T. A. Dowling, Eds.). University of North Carolina Press, Chapel Hill (1969) L. R. Ford and D. R. Fulkerson, Flows in Networks. Princeton University Press, Princeton, NJ (1962). P. Katerinis, Minimum degree of bipartite graphs and the existence of k-factors. Graphs Combinatorics 6 L. Lovhsz and M. D. Plummer, Matching theory. Ann. ofDiscrete Math. 29 (1986). E. A. van Doorn, personal communication. 561-577. (1990) 253-258. Received July 1991 Accepted October 1992 

Edge colorings in bipartite graphs.

Graph Theory with Applications.

Minimum degree of bipartite graphs and the existence of k-factors.

http://purl.utwente.nl/publications/71001

Decomposition of bipartite graphs under degree constraints

Abstract

Similar works

Full text

Available Versions

NARCIS

University of Twente Research Information