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Counting 1-factors in infinite graphs

Ron Aharoni, Mao Lin Zheng
1990 Journal of combinatorial theory. Series B (Print)  
We prove that for n 33 any infinite n-connected factorizable graph has at least n! l-factors (which is a sharp lower bound). ci;l  ...  B 34 (1983). 48-57) proved that a locally finite infinite n-connected factorizable graph has at least (n-l)! l-factors and showed that for n =2 this lower bound is sharp.  ...  Zf G is an infinite matchable and bicritical graph then f(G) is infinite. The following is a generalization of a result in [6] : THEOREM 5.3 .  ... 
doi:10.1016/0095-8956(90)90072-8 fatcat:fjrtgtv44rawxkhtrvuaomqetq

LP duality in infinite hypergraphs

Ron Aharoni, Ran Ziv
1990 Journal of combinatorial theory. Series B (Print)  
The proof uses a Gallai-Edmonds decomposition result for infinite graphs.  ...  In any graph there exist a fractional cover and a fractional matching satisfying the complementary slackness conditions of linear programming.  ...  A graph G is matchable if and only if n(G, S) is espousable for every S c V(G). Q -a.  ... 
doi:10.1016/0095-8956(90)90098-k fatcat:uweawdnfxrcqjachp7rnhpfyai

Infinite matching theory

Ron Aharoni
1991 Discrete Mathematics  
Some results are presented which have not appeared elsewhere, mainly concerning Menger's theorem for infinite graphs.  ...  We survey the existing theory of matchings in infinite graphs and hypergraphs, with special attention to the duality between matchings and covers.  ...  We can now state Gallai's theorem in a form which is true also for infinite graphs. Theorem 5.1 [6]. A graph G is matchable if and only if l7(G, S) is espousable for every set of vertices S.  ... 
doi:10.1016/0012-365x(91)90327-x fatcat:erhec3sitzevzextr4xb6skt34

Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas

Ron Aharoni, Nathan Linial
1986 Journal of combinatorial theory. Series A  
In any bipartite graph r= (X, Y, K) there exists a cover C = A u B, where A c X and B E Y, such that A is matchable into Y\B and B is matchable into X\A.  ...  This is easily seen to be equivalent to a version which was proved in [I] to hold also for infinite graphs: THEOREM K.  ... 
doi:10.1016/0097-3165(86)90060-9 fatcat:l744d75wwzdibg5opsp74fsire

A compactness theorem for perfect matchings in matroids

P Komjath, E.C Milner, N Polat
1988 Journal of combinatorial theory. Series B (Print)  
Thus, in our terminology, if & is the set of circuits of a graph (the elements of a member E E 6 are the edges comprising E), then d is matchable iff B is finitely matchable.  ...  Suppose for contradiction that I is infinite.  ... 
doi:10.1016/0095-8956(88)90035-4 fatcat:u546twqtu5cgrm6igegzpptnk4

Up to a double cover, every regular connected graph is isomorphic to a Schreier graph [article]

Paul-Henry Leemann
2022 arXiv   pre-print
We prove that every connected locally finite regular graph has a double cover which is isomorphic to a Schreier graph.  ...  This result extends by compacity to infinite 2d-regular connected graph without degenerated loop, see [3] for a proof.  ...  Observe that not every regular graphs of odd degree are matchable, even among graphs without degenerated loop. See Figure 2 .  ... 
arXiv:2010.06431v2 fatcat:dya33jxqlffmrb7gbzbenymhgy

On Perfect Matchings in Matching Covered Graphs [article]

Jinghua He, Erling Wei, Dong Ye, Shaohui Zhai
2017 arXiv   pre-print
In this paper, we show that, for every integer k> 3, there exist infinitely many k-regular graphs of class 1 with an arbitrarily large equivalent class K such that K is not switching-equivalent to either  ...  Further, we characterize bipartite graphs with equivalent class, and characterize matching-covered bipartite graphs of which every edge is removable.  ...  Let G(A, B) be a matchable bipartite graph.  ... 
arXiv:1703.05412v2 fatcat:yunkqjonozbszm37ttxsuunzt4

Strong LP duality in weighted infinite bipartite graphs

Ron Aharoni, Vladimir Korman
1994 Discrete Mathematics  
We prove a weighted generalization of Kiinig's duality theorem for infinite bipartite graphs and a weighted version of its dual. R. Aharoni. V. Korman/Discrete Mathematics 131 (1994) 1-7  ...  Not every infinite weighted bipartite graph has an orthogonal pair (matching, w-cover).  ...  Hence, if A is infinite, r, cannot be inespousable for more than IAl many values of p. Thus, for some ordinal 0< [Al+, the graph r, is espousable. EE(&).  ... 
doi:10.1016/0012-365x(94)90367-0 fatcat:hkc4osoov5c5nh4fonstysahpe

A generalization of Tutte's 1-factor theorem to countable graphs

Ron Aharoni
1984 Journal of combinatorial theory. Series B (Print)  
A criterion is proved for a countable graph to possess a perfect matching, in terms of "marriage" in bipartite graphs associated with the graph.  ...  The criterion is conjectured to be valid for general graphs.  ...  A graph G is called "peculiar" ("factor-critical" in the terminology of [7] ) if it is unmatchable, but G -{x} is matchable for every x E V(G).  ... 
doi:10.1016/0095-8956(84)90052-2 fatcat:5nqwfzj5qfgirheagetltt4abi

Page 763 of Mathematical Reviews Vol. 51, Issue 3 [page]

1976 Mathematical Reviews  
If m is the maximum number of edges in a matching of a graph G having n nodes then 2m/n is called the matchability ug, of G. The matchability ys of a class S of graphs is defined as glbges we.  ...  The essential matchability »* of S is defined as lub, u(S,), where S,, is the set of graphs in S having at least kK nodes.  ... 

Matchings in infinite graphs

Ron Aharoni
1988 Journal of combinatorial theory. Series B (Print)  
A general criterion is proved for a graph of any cardinality to possess a perfect matching. The criterion is used to prove an extension of Tutte's l-factor theorem for general graphs.  ...  In [S] a criterion for matchability of one side of a bipartite graph was obtained for graphs of any cardinality.  ...  The graph GA -{x} contains a subgraph isomorphic to (G -{xi)"' c-Xi as the union of connected components, and hence this last graph is also matchable.  ... 
doi:10.1016/0095-8956(88)90098-6 fatcat:u6k7jpm6y5futdpeaze27eljyq

Arithmetically maximal independent sets in infinite graphs

Stanisław Bylka
2005 Discussiones Mathematicae Graph Theory  
A positive answer is given to the following classes of infinite graphs: bipartite graphs, line graphs and graphs having locally infinite clique-cover of vertices. Some counter examples are presented.  ...  Families of all sets of independent vertices in graphs are investigated. The problem how to characterize those infinite graphs which have arithmetically maximal independent sets is posed.  ...  Therefore, Theorem 4.3 may be generalized to all line graphs of multigraphs which possess maximal matchable subsets of vertices -for example, the line graphs of multigraphs without infinite paths.  ... 
doi:10.7151/dmgt.1270 fatcat:xbzyjjmklrbedcvegorn24q63a

Conditions for matchability in groups and field extensions [article]

Mohsen Aliabadi, Jack Kinseth, Christopher Kunz, Haris Serdarevic, Cole Wills
2022 arXiv   pre-print
Note that the method of associating a bipartite graph to our subsets in Theorem 1.1 first was used in [1] as a tool to count the number of matchings of matchable subsets of a given abelian group.  ...  Assume that K is infinite and K ⊂ F is simple.  ... 
arXiv:2107.09029v3 fatcat:qi5gmi6a5jehbn5av5r3kwfegu

Strongly maximal matchings in infinite weighted graphs [article]

Ron Aharoni, Eli Berger, Agelos Georgakopoulos, Philipp Sprüssel
2009 arXiv   pre-print
Given an assignment of weights w to the edges of a graph G, a matching M in G is called strongly w-maximal if for any matching N the sum of weights of the edges in N\M is at most the sum of weights of  ...  A similar situation occurs when studying matchings in infinite graphs.  ...  A graph C is called almost matchable if C − v has a perfect matching for some v ∈ V (C). It is called uniformly almost matchable if C − v has a perfect matching for every v ∈ V (C).  ... 
arXiv:0911.4010v1 fatcat:yexxxciyvvcx5kpxxxs5lhzluy

Strongly Maximal Matchings in Infinite Graphs

Ron Aharoni, Eli Berger, Agelos Georgakopoulos, Philipp Sprüssel
2008 Electronic Journal of Combinatorics  
Given an assignment of weights $w$ to the edges of an infinite graph $G$, a matching $M$ in $G$ is called strongly $w$-maximal if for any matching $N$ there holds $\sum\{w(e) \mid e \in N \setminus M\}  ...  This result is best possible in the sense that if we allow irrational values or infinitely many values then there need not be a strongly $w$-maximal matching.  ...  A graph C is called almost matchable if C−v has a perfect matching for some v ∈ V (C). It is called uniformly almost matchable if C −v has a perfect matching for every v ∈ V (C).  ... 
doi:10.37236/860 fatcat:pn4rl3ut6nbv3ld5xn6scsuppe
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