Bicolored Matchings in Some Classes of Graphs

We consider the problem of finding in a graph a set R of edges to be colored in red so that there are maximum matchings having some prescribed numbers of red edges. For regular bipartite graphs with n nodes on each side, we give sufficient conditions for the existence of a set R with |R|=n+1 such that perfect matchings with k red edges exist for all k,0≤k≤n. Given two integers p<q we also determine the minimum cardinality of a set R of red edges such that there are perfect matchings with p red edges and with q red edges. For 3-regular bipartite graphs, we show that if p≤4 there is a set R with |R|=p for which perfect matchings M_k exist with |M_k∩R|≤k for all k≤p. For trees we design a linear time algorithm to determine a minimum set R of red edges such that there exist maximum matchings with k red edges for the largest possible number of values of k.     

On Perfect k-Matchings

In this paper, we generalize the notions of perfect matchings, perfect 2-matchings to perfect k-matchings and give a necessary and sufficient condition for the existence of perfect k-matchings. We show that a bipartite graph G contains a perfect k-matching if and only if it contains a perfect matching. Moreover, for regular graphs, we provide a sufficient condition for the existence of perfect k-matching in terms of the edge connectivity.     

Perfect Matchings of Regular Bipartite Graphs

Let G be a regular bipartite graph and . We show that there exist perfect matchings of G containing both, an odd and an even number of edges from X if and only if the signed graph , that is a graph G with exactly the edges from X being negative, is not equivalent to . In fact, we prove that for a given signed regular bipartite graph with minimum signature, it is possible to find perfect matchings that contain exactly no negative edges or an arbitrary one preselected negative edge. Moreover, if the underlying graph is cubic, there exists a perfect matching with exactly two preselected negative edges. As an application of our results we show that each signed regular bipartite graph that contains an unbalanced circuit has a 2‐cycle‐cover such that each cycle contains an odd number of negative edges.     

On zero-sum -flows of graphs

For a graph G, a zero-sum flow is an assignment of non-zero real numbers on the edges of G such that the total sum of all edges incident with any vertex of G is zero. A zero-sum k-flow for a graph G is a zero-sum flow with labels from the set {±1,…,±(k-1)}. In this paper for a graph G, a necessary and sufficient condition for the existence of zero-sum flow is given. We conjecture that if G is a graph with a zero-sum flow, then G has a zero-sum 6-flow. It is shown that the conjecture is true for 2-edge connected bipartite graphs, and every r-regular graph with r even, r>2, or r=3.     

A note on 1-factors in graphs

Conditions on a graphG are presented which are sufficient to guarantee thatG–Z contains a 1-factor, whereZ is a set of edges ofG of restricted cardinality. These conditions provide generalizations of several known results and, further, establish the result that ifG is anr-regular, (r–2)-edge-connected graph (r2) of even order andz is an integer with 0zr–1 such thatG contains fewer thanr–z edge cut sets of cardinalityr–2, thenG–Z has a 1-factor for each setZ ofz edges ofG.     

Perfect matchings in uniform hypergraphs with large minimum degree

A perfect matching in a k-uniform hypergraph on n vertices, n divisible by k, is a set of n/k disjoint edges. In this paper we give a sufficient condition for the existence of a perfect matching in terms of a variant of the minimum degree. We prove that for every k≥3 and sufficiently large n, a perfect matching exists in every n-vertex k-uniform hypergraph in which each set of k−1 vertices is contained in n/2+Ω(logn) edges. Owing to a construction in [D. Kühn, D. Osthus, Matchings in hypergraphs of large minimum degree, J. Graph Theory 51 (1) (2006) 269–280], this is nearly optimal. For almost perfect and fractional perfect matchings we show that analogous thresholds are close to n/k rather than n/2.     

Edge-Coloring Bipartite Graphs

Given a bipartite graph G with n nodes, m edges, and maximum degree Δ, we find an edge-coloring for G using Δ colors in time T + O(m log Δ), where T is the time needed to find a perfect matching in a k-regular bipartite graph with O(m) edges and k ≤ Δ. Together with best known bounds for T this implies on edge-coloring algorithm which improves on the algorithm of Hopcroft and Cole. Our algorithm can also be used to find a (Δ + 2)-edge-coloring for G in time O(m log Δ). The previous best approximation algorithm with the same time bound needed Δ + log Δ colors.     

<Emphasis Type="Italic">k</Emphasis>-factors in regular graphs

Plesnik in 1972 proved that an （m - 1）-edge connected m-regular graph of even order has a 1-factor containing any given edge and has another 1-factor excluding any given m - 1 edges. Alder et al. in 1999 showed that if G is a regular （2n ＋ 1）-edge-connected bipartite graph, then G has a 1-factor containing any given edge and excluding any given matching of size n. In this paper we obtain some sufficient conditions related to the edge-connectivity for an n-regular graph to have a k-factor containing a set of edges and （or） excluding a set of edges, where 1 ≤ k ≤n/2. In particular, we generalize Plesnik＇s result and the results obtained by Liu et al. in 1998, and improve Katerinis＇ result obtained 1993. Furthermore, we show that the results in this paper are the best possible.     

Matching graphs of hypercubes and complete bipartite graphs

Kreweras’ conjecture [G. Kreweras, Matchings and hamiltonian cycles on hypercubes, Bull. Inst. Combin. Appl. 16 (1996) 87–91] asserts that every perfect matching of the hypercube Q_d can be extended to a Hamiltonian cycle of Q_d. We [Jiří Fink, Perfect matchings extend to hamilton cycles in hypercubes, J. Combin. Theory Ser. B, 97 (6) (2007) 1074–1076] proved this conjecture but here we present a simplified proof.The matching graph of a graph G has a vertex set of all perfect matchings of G, with two vertices being adjacent whenever the union of the corresponding perfect matchings forms a Hamiltonian cycle of G. We show that the matching graph of a complete bipartite graph is bipartite if and only if n is even or n=1. We prove that is connected for n even and has two components for n odd, n≥3. We also compute distances between perfect matchings in .     

Strong chromatic index of subset graphs

The strong chromatic index of a graph G, denoted sq(G), is the minimum number of parts needed to partition the edges of G into induced matchings. For 0 ≤ k ≤ l ≤ m, the subset graph S_m(k, l) is a bipartite graph whose vertices are the k- and l-subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. We show that and that this number satisfies the strong chromatic index conjecture by Brualdi and Quinn for bipartite graphs. Further, we demonstrate that the conjecture is also valid for a more general family of bipartite graphs. © 1997 John Wiley & Sons, Inc.     

Transversal hypergraphs to perfect matchings in bipartite graphs: Characterization and generation algorithms

A minimal blocker in a bipartite graph G is a minimal set of edges the removal of which leaves no perfect matching in G. We give an explicit characterization of the minimal blockers of a bipartite graph G. This result allows us to obtain a polynomial delay algorithm for finding all minimal blockers of a given bipartite graph. Equivalently, we obtain a polynomial delay algorithm for listing the anti‐vertices of the perfect matching polytope of G. We also provide generation algorithms for other related problems, including d‐factors in bipartite graphs, and perfect 2‐matchings in general graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 209–232, 2006     

Connected matchings in chordal bipartite graphs

A connected matching in a graph is a collection of edges that are pairwise disjoint but joined by another edge of the graph. Motivated by applications to Hadwiger’s conjecture, Plummer, Stiebitz, and Toft (2003) introduced connected matchings and proved that, given a positive integer $k$, determining whether a graph has a connected matching of size at least $k$ is NP-complete. Cameron (2003) proved that this problem remains NP-complete on bipartite graphs, but can be solved in polynomial-time on chordal graphs. We present a polynomial-time algorithm that finds a maximum connected matching in a chordal bipartite graph. This includes a novel edge-without-vertex-elimination ordering of independent interest. We give several applications of the algorithm, including computing the Hadwiger number of a chordal bipartite graph, solving the unit-time bipartite margin-shop scheduling problem in the case in which the bipartite complement of the precedence graph is chordal bipartite, and determining–in a totally balanced binary matrix–the largest size of a square sub-matrix that is permutation equivalent to a matrix with all zero entries above the main diagonal.     

一类变换图的连通性

本文定义了一类由给定的一个３－正则平面偶图的全体完美匹配所构成的变换图，并证明了该变换图是连通的．由此可得出结论：从任一给定的３－正则平面偶图的完美匹配出发，通过一种所谓的旋转运算，就可以生成全部其它的完美匹配．     

Forcing faces in plane bipartite graphs

Let Ω denote the class of connected plane bipartite graphs with no pendant edges. A finite face s of a graph G∈Ω is said to be a forcing face of G if the subgraph of G obtained by deleting all vertices of s together with their incident edges has exactly one perfect matching. This is a natural generalization of the concept of forcing hexagons in a hexagonal system introduced in Che and Chen [Forcing hexagons in hexagonal systems, MATCH Commun. Math. Comput. Chem. 56 (3) (2006) 649-668]. We prove that any connected plane bipartite graph with a forcing face is elementary. We also show that for any integers n and k with n?4 and n?k?0, there exists a plane elementary bipartite graph such that exactly k of the n finite faces of G are forcing. We then give a shorter proof for a recent result that a connected cubic plane bipartite graph G has at least two disjoint M-resonant faces for any perfect matching M of G, which is a main theorem in the paper [S. Bau, M.A. Henning, Matching transformation graphs of cubic bipartite plane graphs, Discrete Math. 262 (2003) 27-36]. As a corollary, any connected cubic plane bipartite graph has no forcing faces. Using the tool of Z-transformation graphs developed by Zhang et al. [Z-transformation graphs of perfect matchings of hexagonal systems, Discrete Math. 72 (1988) 405-415; Plane elementary bipartite graphs, Discrete Appl. Math. 105 (2000) 291-311], we characterize the plane elementary bipartite graphs whose finite faces are all forcing. We also obtain a necessary and sufficient condition for a finite face in a plane elementary bipartite graph to be forcing, which enables us to investigate the relationship between the existence of a forcing edge and the existence of a forcing face in a plane elementary bipartite graph, and find out that the former implies the latter but not vice versa. Moreover, we characterize the plane bipartite graphs that can be turned to have all finite faces forcing by subdivisions.     

Triangle-free graphs with uniquely restricted maximum matchings and their corresponding greedoids

A matching M is uniquely restricted in a graph G if its saturated vertices induce a subgraph which has a unique perfect matching, namely M itself [M.C. Golumbic, T. Hirst, M. Lewenstein, Uniquely restricted matchings, Algorithmica 31 (2001) 139-154]. G is a König-Egerváry graph provided α(G)+μ(G)=|V(G)| [R.W. Deming, Independence numbers of graphs—an extension of the König-Egerváry theorem, Discrete Math. 27 (1979) 23-33; F. Sterboul, A characterization of the graphs in which the transversal number equals the matching number, J. Combin. Theory Ser. B 27 (1979) 228-229], where μ(G) is the size of a maximum matching and α(G) is the cardinality of a maximum stable set. S is a local maximum stable set of G, and we write S∈Ψ(G), if S is a maximum stable set of the subgraph spanned by S∪N(S), where N(S) is the neighborhood of S. Nemhauser and Trotter [Vertex packings: structural properties and algorithms, Math. Programming 8 (1975) 232-248], proved that any S∈Ψ(G) is a subset of a maximum stable set of G. In [V.E. Levit, E. Mandrescu, Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings, Discrete Appl. Math. 132 (2003) 163-174] we have proved that for a bipartite graph G,Ψ(G) is a greedoid on its vertex set if and only if all its maximum matchings are uniquely restricted. In this paper we demonstrate that if G is a triangle-free graph, then Ψ(G) is a greedoid if and only if all its maximum matchings are uniquely restricted and for any S∈Ψ(G), the subgraph spanned by S∪N(S) is a König-Egerváry graph.     

On matching and total domination in graphs

A set M of edges of a graph G is a matching if no two edges in M are incident to the same vertex. A set S of vertices in G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The matching number is the maximum cardinality of a matching of G, while the total domination number of G is the minimum cardinality of a total dominating set of G. In this paper, we investigate the relationships between the matching and total domination number of a graph. We observe that the total domination number of every claw-free graph with minimum degree at least three is bounded above by its matching number, and we show that every k-regular graph with k?3 has total domination number at most its matching number. In general, we show that no minimum degree is sufficient to guarantee that the matching number and total domination number are comparable.     

Perfect Matchings in ε-Regular Graphs and the Blow-Up Lemma

G on vertex set , , with density d>2ε and all vertex degrees not too far from d, has about as many perfect matchings as a corresponding random bipartite graph, i.e. about . In this paper we utilize that result to prove that with probability quickly approaching one, a perfect matching drawn randomly from G is spread evenly, in the sense that for any large subsets of vertices and , the number of edges of the matching spanned between S and T is close to |S||T|/n (c.f. Lemma 1). As an application we give an alternative proof of the Blow-up Lemma of Komlós, Sárk?zy and Szemerédi [10]. Received: December 5, 1997     

Hamilton Decomposable Graphs with Specified Leaves

In this paper, a simple proof is given of a result that provides necessary and sufficient conditions for the existence of a hamilton decomposition of G–E(H) for any non-bipartite r-regular complete multipartite graph G and for any 2-factor H of G. Such conditions were originally obtained by Buchanan for complete graphs (ie when r=|V(G)|–1), and in some cases by Leach and Rodger otherwise (Leach and Rodger also settled the bipartite case). This result is extended to consider hamilton decompositions of G–E(HF), where F is a 1-factor of G.     

The rainbow number of matchings in regular bipartite graphs

Given a graph G and a subgraph H of G, let rb(G,H) be the minimum number r for which any edge-coloring of G with r colors has a rainbow subgraph H. The number rb(G,H) is called the rainbow number of H with respect to G. Denote as mK₂ a matching of size m and as B_n,k the set of all the k-regular bipartite graphs with bipartition (X,Y) such that X=Y=n and k≤n. Let k,m,n be given positive integers, where k≥3, m≥2 and n>3(m−1). We show that for every GB_n,k, rb(G,mK₂)=k(m−2)+2. We also determine the rainbow numbers of matchings in paths and cycles.