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					Combinatorial Optimization Hangzhou, May 2004 ·

Problems on the . . . Some problems on . . .

Some Problems on Combinatorial Optimization in Fractional Graph Theory

Fractional Hamiltonian . . . Fractional (g, f )- . . . Fractional coloring . . . Computational complexity

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Liu Guizhen
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Department of Mathematics Shandong University gzliu@sdu.edu.cn or gzliu@math.sdu.edu.cn

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Problems on the . . . Some problems on . . . Fractional Hamiltonian . . . Fractional (g, f )- . . .

Our purpose is to reveal the rational side of graph theory. we seek to convert integer-based definitions and invariants into their fractional analogues. Find the relationships of the programming and the graph theory. Some problems on combinatorial optimization in fractional graph Theory are introduced. In particular, many results on fractional factors and fractional colorings are presented. Furthermore, some open problems are presented. In the following we give some basic concepts and results on hypergraphs which are very important for our problems on combinatorial optimization.

Fractional coloring . . . Computational complexity

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Problems on the fractional covering

Fractional Hamiltonian . . . Fractional (g, f )- . . . Fractional coloring . . .

and packing of hypergraphs
• A hypergraph H is a pair (S, X), where S is a finite set and X is a family of subsets of S. The set S is called the vertex set of the hypergraph, and so we sometimes write V (H) for S. The elements of X are called huperedges or sometimes just edges.

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A covering of H is a collection of hyperedges X1 , X2 , · · · , Xj so that S ⊂ X1 ∪ X2 ∪ · · · Xj . The least j for which this is possible is called the covering number of H, and is denoted by k(H). The covering problem can be formulated as an integer program (IP). To each set Xi ∈ X associate a 0, 1-variable xi . The vector x is a indicator of the sets we have selected for the cover. Let M be the vertex-hyperedge incidence matrix of H. The condition that the indicator vector x corresponds to a covering is simply M x ≥ 1 ( that is, every coordinate of M x is at least 1) Thus k(H) is the value of the integer program minimize I x subject to M x ≥ 1 and x ≥ 0 where I represents a vector of all ones.

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• A packing of a hypergraph H is a subset Y ⊂ S with the property that no two elements of Y are together in the same number of X. The packing number p(H) is defined to be the largest size of a packing. The packing number of a graph is its independence number. There is a corresponding IP formulation. Let yi be a 0, 1-indicator variable that is 1 just when si ∈ Y. The condition that Y is a packing is simply M y ≤ 1 where M is as above. Thus p(H) is the value of the integer program maximize I y subject to M yx ≤ 1 and y ≥ 0. This is the dual IP to the covering problem and the following result is true.

Fractional Hamiltonian . . . Fractional (g, f )- . . . Fractional coloring . . . Computational complexity

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• 1.1 For a hypergraph H, we have p(H) ≤ k(H). Note that many graph theory concepts can be seen as hypergraph covering or packing problems. For instance, the chromatic number and matching number. Now we consider fractional covering and packing. We define the fractional covering number and fractional packing number of H, denoted by kf (H) and pf (H) respectively, to be the values of the dual linear program “ minimize I x subject to M x ≥ 1 and x ≥ 0” and “maximize I y subject to M yx ≤ 1 and y ≥ 0.” By duality we have kf (H) = pf (H).

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There is a second way to define the fractional covering and packing numbers. We begin by define t−fold covering and t−fold covering number of H where t is a positive integer. A t-fold covering of H is a multiset {X1 , X2 , · · · , Xj } where Xi ∈ X with the property that each s ∈ S is in at least t of the Xi s. The smallest cardinality (least j) of such multiset is called the t-fold covering number of H and denoted by kt (H). Clearly, k1 (H) = k(H). Note that ks+t (H) ≤ ks (H) + kt (H). Therefor we define the fractional covering number of H to be kf (H) = lim kt (H) kt (H) = inf . t−→∞ t t t

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We can prove that the above two definitions are the same and kf (H) ≤ k(H).

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In the same way, we define a t-fold packing of H to be a multiset Y of the vertex set with the property that for every Xi ∈ X we have s∈X m(s) ≤ t where m is the multiplicity of s ∈ S in Y. The t−fold packing number of H, denoted by pt (H), is the largest cardinality of a t-fold packing. Observe that p1 (H) = p(H). We define the fractional packing number of H to be pf (H) = lim pt (H) . t−→∞ t

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We also have pf (H) ≥ k(H). It is easy to see that the following result holds. 1.2 If H has no exposed vertices, then kf (H) is a rational number and there exists a positive integer s for which kf (H) = ks (H)/s.
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ings

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• A fractional matching is a function h that assigns to each edge of a graph G a number in [0, 1] so that, for each vertex x , we have h(e) ≤ 1 where the sum is taken over all edges incident to x. If h(e) ∈ {0, 1} for every edge e, then f is just a matching, or more precisely, the indicator function of a matching. • The fractional matching number µh (G) of a graph G is the supremum of h(e) over all fractional matchings h.
e∈E(G)

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On the other hand, given a graph G we can construct a hypergraph Hwhose ground vertex set is E(G) and with a hyperedge ev for each vertex v ∈ V (G) consisting of all edges in E(G) incident to v. It is easy to see that the matching number of G satisfies µ(G) = p(H) and the fractional matching number µh (G) = pf (H).

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It is clear that µh (G) ≥ µ(G). The following results are based on the work of Balas, Balinski, Bourjolly, Lov´ sz, plummuer, Pulleyblank, and Uhry and so on. a • 2.1. µh (G) ≤ 1 |V (G)|. 2 • 2.2. If G is bipartite, Then µh (G) = µ(G). • 2.3. For any graph G, 2µh (G) is an integer. Moreover, there is a fractional matching h for which h(e) = µh (G)
e∈E(G)

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such that h(e) ∈ {0, 1/2, 1} for every edge e.

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• A fractional transversal of graph G is a function p : V (G) → [0, 1] satisfying p(x) ≥ 1 for every e ∈ E(G) which is the dual concept of fractional
x∈e

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matching. . The fractional transversal number is the infimum of
x∈V (G)

p(x)

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taken over all fractional transversals p of G. • 2.4. For every graph G, there is a fractional transversal p for which p(x) = µh (G)
x∈V (G)

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such that p(x) ∈ {0, 1/2, 1} for every vertex x. . Suppose that h is a fractional matching. Then h is a fractional 1-factor(or perfect matching) if and only if for every x ∈ V (G), h(e) = 1.
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• 2.5. Fractional Tutte’s Theorem A graph G has a fractional 1-factor (or perfect matching) if and only if i(G − S) ≤ |S| for every set S ⊆ V (G) where i(G − S) is the number of isolated vertices of G − S. • 2.6. Fractional Berge’s Theorem For any graph G, 1 µh (G) = (|V (G)| − max{i(G − S) − |S|}) 2 where the maximum is taken over all S ⊆ V (G).

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The above results can be proved by graph theory method or programming method and they are basic for our fractional factor theory. Now we give some new results on fractional matching which obtained by us. At first we introduce two concepts. • The isolated toughness of G is defined as

Problems on the . . . Some problems on . . . Fractional Hamiltonian . . . Fractional (g, f )- . . . Fractional coloring . . . Computational complexity

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i(G) = min{

|S| : S ⊂ V (G), i(G − S) ≥ 2}. i(G − S)

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if G is not complete. Otherwise, i(G) = ∞. • If graph G has a k-matching (a matching with k edges) and for any kmatching M graph G has a fractional 1-factor containing M, then we see that G is fractional k-extendable.
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• 2.7 (Ma 2002) Let G be a graph and k be an non-negative integer. If the connectivity κ(G) ≥ 2k + 1, i(G) ≥ k + 1 and |V (G)| ≥ 2k + 1, then G is fractional k-extendable. • 2.8 (Ma and Liu 2003) Let G be a graph with |V (G)| ≥ 2k + 4 and let F be an arbitrary 1-factor of G. If G − uv is fractional k-extendable for each e = uv ∈ F, then G is fractional k-extendable. • 2.9 (Yu and Liu 2003) A graph G has a fractional 1-factor if and only if bind(G) ≥ 1. • 2.10. (Liu Yan and Liu Guizhen 2002) Let G be a graph. Then µh (G) = µ(G) if and only if D(G) is an independent set where D(G) is the set of all vertices in G which are missed by at least one maximum matching of G.

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Finally we present the following problems. • Problem 2.1 Find the relationship between binding number and fractional k-extendable of a graph. • Problem 2.2 Give an algorithm for determining whether a graph is fractional k-extendable.

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Fractional Hamiltonian Problems

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A graph G is called fractionally Hamiltonian (F H) if there is a function h : E(G) → [0, 1] such that the following two conditions holds h(e) = |V (G)|
e∈E(G)

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and for all ∅ ⊂ S ⊂ V (G) h(e) ≥ 2
¯ e∈[S,S]

¯ where [S, S] for the set of all edges with exactly one end in S. We called such a function h a fractional Hamiltonian cycle. For example, Petersen’s graph is F H , but is not Hamiltonian.

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• 3.1. Let h be a fractional Hamiltonian cycle for a graph G and let x be any vertex of G. Then h(e) = 2 .
e x
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• 3.2. If G is F H, then G has a fractional 1-factor. • 3.3. If G is F H and |V (G) ≥ 3 , then t(G) ≥ 1. • 3.4. Let k < 3/2. There is a graph G with t(G) ≥ k that is not F H. • F HLP min subject to
¯ e∈[S,S]

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we ≥ 2 we ≥ 0,

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• 3.5. Let G be a graph on at least 3 vertices. Then G is F H iff the value of the F HLP is exactly |V (G)|. • 3.6. Let t < 3 . Then there is a graph G with t(G) ≥ t that is not fractional 2 Hamiltonian.

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Note that there is a graph G with t(G) ≥ 2 that is not Hamiltonian. But the following conjecture is presented in [5]. • Conjecture 3.1. If t(G) ≥ 2, then G is fractional Hamiltonian. There are many problems on graphs which is fractional Hamiltonian can be considered. For example, we can consider the structures and properties of fractional Hamiltonian graphs. • Problem 3.2 Find the relationship between binding number and fractional Hamiltonian in graphs. 3.1. Let h be a fractional Hamiltonian cycle for a graph G and let x be any vertex of G. Then h(e) = 2
e x

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Fractional (g, f )-factor problems
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Let G be a graph and let g and f be two integer-valued functions defined on V (G) such that 0 ≤ g(x) ≤ f (x) for all x in V (G). Let dh (x) = h(e). G Afractional (g, f )-factor is a function h that assigns to each edge of a graph a number in [0, 1] so that, for each vertex x of G we have g(x) ≤ dh (x) ≤ f (x). G If g(x) = f (x) for all x ∈ V (G), then a fractional (g, f )-factor is called an fractional f -factor. Let a and b be two non-negative integers. If g(x) = a and f (x) = b for every x ∈ V (G), then a fractional (g, f )-factor is called a fractional [a, b]-factor. If a = b = k, an fractional [a, b]-factor is called a fractional k-factor. It is easy to see that a fractional (0, 1)-factor is a fractional matching.

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• 4.1. Fractional Lov´ sz’s theorem a A graph G has a fractional (g, f )-factor iff for any subset S of V (G) g(T ) − dG−S (T ) ≤ f (S)
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where T = {x : x ∈ V (G) \ S, dG−S (x) ≤ g(x)}. ( obtained by Astee in 1990) • 4.2 (Zhang and Liu 1999) A graph G has a fractional k-factor iff for any subset S of V (G)
k−1

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where pj (G − S) denotes the number of vertices in G − S with degree j. • (Liu and Zhang 1999) Let G be a bipartite graph or g(x) = f (x) for all x ∈ V (G). Then G has a fractional (g, f )-factor iff G has a (g, f )-factor. 3.1. Let h be a fractional Hamiltonian cycle for a graph G and let x be any vertex of G. Then h(e) = 2 .
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• 4.4. (Liu 1999) Let G ba a graph. If for every pairs of vertices x , y of G, g(y)dG (x) ≤ f (x)dG (y),
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then G has a fractional (g, f )-factor. • 4.5 (Liu 1999) There are polynomial algorithms for finding fractional (g, f )factor and a maximum fractional (g, f )-factor in graphs. • 4.6 (Ma and Liu 2003) Let G be a connected graph and k > 0 be an integer. If δ(G) ≥ k and i(G) ≥ k, then G has a fractional k-factor. • 4.7 (Ma and Liu 2002) Let G be a graph and let a and b be two integers such that a < b. If i(G) ≤ a − 1 + a and δ(G) ≥ i(G), then G has an fractional b (a, b)-factor.

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• 4.8 (Liu and Zhang 2003) Let G be a graph with |V (G)| ≥ k + 1 where 1 k ≥ 2. Then G has a fractional k-factor if t(G) ≥ k − k . • 4.9(Liu and Zhang 2003) If graph G has a fractional (g, f )-factor, then G 1 has a factional (g, f )-factor h such that h(e) ∈ {0, 1, 2 }.

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We have found the polynomial algorithms for finding the maximum fractional (g,f)-factors by finding the increasing chains in a graph. Finally we give the following conjecture and problems. • Conjecture 4.1. Let G be a graph. If t(G) ≥ k, then G has a connected fractional k-factor. • Problem 4.1. Find the relationship between fractional factors and the factors in graphs. • Problem 4.2. Find the relationship between fractional factors and the binding number of a graph. • Problem 4.3. Find the sufficient conditions for a graph to have connected fractional (g, f )-factors. • Problem 4.4. Describe and solve the above problems by linear program method.

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Fractional coloring Problems
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The fractional chromatic number is defined as follows. A b-fold coloring of a graph G assigns to each vertex of G a set of b colors so that adjacent vertices receive disjoint sets of colors. We say that G is a : b-colorable if it has a b-fold coloring in which the colors are drawn from a palette of a colors. We also call this coloring an a : b-coloring. The least a for which G has a b-fold coloring is the b-fold coloring number of G, denoted χb (G). Note that χ1 (G) = χ(G). Thus the fractional chromatic number to be χb (G) χb (G) = inf . χf (G) = lim b b−→∞ b b On the other hand we can describe the fractional chromatic number as follows. Given graph G, we construct a hypergraph H such that the vertex set of H is the vertex set of G and the hyperedges of H are the independent sets of G. Then it ia easy to see that k(H) = χ(G), χf (G) = kf (G) and χf (G) = χ(G). Note that perfection is trivial in fractional graph theory. We have the following result and problem.

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• 5.1.The fractional chromatic number of a graph equals its fractional list number. • Problem 5.1 when we have χf (G) = χ(G)? • Problem 5.2 Given graph G how to find χf (G)? Note that computing χf can not be done in polynomial time although the linear program can be solve by polynomial time. This is because of the linear program may has exponentially many ( in the number of vertices) variables. One for each maximal independent set in the graph. 4.1. Fractional Lov´ sz’s theorem a

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Computational complexity

Fractional coloring . . . Computational complexity

Many problems on fractional graph theory can be solved in polynomial time. The constraint matrix in fractional matching number problem has size |V (G)| × |E(G)|. Thus polynomial linear programming solutions for this problem exist. Therefore the fractional matching number of a graph can be computed in polynomial time.

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The decision problem of determining whether a graph is Hamiltonian is N P complete. But the fractional Hamiltonicity can be tested in polynomial time. We can formulate fractional Hamiltonicity as a multicommodity flow problem and solve it by polynomial algorithm. On the other hand we can solve this problem using linear programming. We assign a weight to each edge of G and seek to minimize the sum of these weights subject to the constraints that the sum of the weights across any edge cut is at least 2. The value of the fractional Hamiltonian linear programming can be found in polynomial time. It is not difficult to see that the fractional (g, f )-factor problem can be formulated as a linear programming and can also be solved in polynomial time. In another way the polynomial algorithms for solving the problem of fractional (g,f)-factors by finding the increasing chains in a graph is given by us. But for fractional coloring problems we have known that it can not be solved by algorithms of polynomial time. It is proved that it ia also NP-complete.

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References
1. Tutte W T. The Factors of Graphs [J]. Canad. J. Math.,1952,4:314-328. 2. Lov´ sz L. Subgraphs with Prescribed Valencies [J]. J. Combin. Theory, 1970, 8:391-416. a 3. Heinrik K, Hell P, Kirpatrick D G and Liu G. A simple existence criterion for (g < f )-factors [J]. Discrete Math., 1990, 85: 313-317. 4. Akiyama J and Kano M. Factors and factorizations of graphs — a survey [J]. J. Graph Theory, 1985, 9: 1-42. 5. Scheinerman Edward R and Ullman Daniel H. Fractional Graph Theory [M]. New York: John Wiley and Sons,Inc., 1997. 6. K¨ nig D. Graphen und matrizen [J]. Math. Lepot., 1931, 38: 116-119. o 7. Balas E. Integer and fractional matchings [M]. In studies on graphs and discrete programming, Ann. Disc. Math. II. Hansen P ed. Noth-Holland, 1981,1-13. 8. Lov´ sz L and Plummer M. Matching Theory [M]. New York: North-Holland, 1986. a 9. Pulleyblank W R. Fractional matchings and the Edmonds-Gallai theorem [J]. Disc. Apll. Math., 1987, 16: 51-58. 10. Yang Jingbo, Ma Yinghong and Liu Guizhen, Fractional (g, f )-factors in Graphs [J]. Appl. Math. J. Chinese Univ. Ser. A, 2001, 16(4): 385-390. 11. Ma Yinghong. Some results on fractional factors of graphs [D]. Ph. D Thesis. Shandong University, 2002.

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12. Ma Yinghong and Liu Guizhen. Some results on fractional k-extendable graphs [J]. Acta. Engineering and mathmatics, to appear. 13. Yu Jiguo and Liu Guizhen. Binding number and minimum degree conditions for graphs to have fractional factors [J]. J. of shandong University, to appear. 14. Liu Yan and Liu Guizhen. The fractional matching numbers of graphs [J]. Networks, 2002, 40(4):228-231. 15. Duffus D, Sands B and Woodrow R. Lexicographic Matching can not form Hamiltonian cycles [J]. Order, 1988, 5: 149-161. 16. Dejter I. Hamilton cycles and quotients in bipartite graphs, In Graph Theory and Applications to Algorithms and Computer Science [M]. Y. Alavi et al., Wiley, 1985, 189-199. 17. Berge C. Fractional Graph Theory. ISI Lecture Notes 1 [M]. Macmillan of India, 1978. 18. Anstee R P. An Algorithmic proof Tutte’s f -factor theorem [J]. J. Algorithms, 1985, 6:112131. 19. Liu Guizhen and Zhang Lanju. Fractional (g, f )-factors of graphs [J]. Acta Math. Scientia, 2001, 21B(4):541-545. 20. Zhang Lan ju and Liu Guizhen. Fractional k-factors of graphs [J]. J. Sys. Sci. and Math. Scis, (2001), 21(1): 88-92. 21. Ma Yinghong and Liu Guizhen. Isolated toughness and the existence of fractional factors [J]. Acta Math. Appl. Sinica, 2003, 26(1):133-140. 22. Ma Yinghong. Isolated toughness and existence of fractional [a, b]-factors of graphs [J]. Acta Math. Scientia, to appear. 23. Zhang Lanju. Factors and fractional factors of graphs and tree graphs [D]. Ph. D Thesis, Shandong University, 2001.

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Thanks!

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Description: factors and fractional factors of graphs