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Counting 1-factors in infinite graphs

1990
*
Journal of combinatorial theory. Series B (Print)
*

We prove that for n 33 any

doi:10.1016/0095-8956(90)90072-8
fatcat:fjrtgtv44rawxkhtrvuaomqetq
*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 . ...##
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LP duality in infinite hypergraphs

1990
*
Journal of combinatorial theory. Series B (Print)
*

The proof uses a Gallai-Edmonds decomposition result for

doi:10.1016/0095-8956(90)90098-k
fatcat:uweawdnfxrcqjachp7rnhpfyai
*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. ...##
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Infinite matching theory

1991
*
Discrete Mathematics
*

Some results are presented which have not appeared elsewhere, mainly concerning Menger's theorem for

doi:10.1016/0012-365x(91)90327-x
fatcat:erhec3sitzevzextr4xb6skt34
*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. ...##
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Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas

1986
*
Journal of combinatorial theory. Series A
*

In any bipartite

doi:10.1016/0097-3165(86)90060-9
fatcat:l744d75wwzdibg5opsp74fsire
*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. ...##
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A compactness theorem for perfect matchings in matroids

1988
*
Journal of combinatorial theory. Series B (Print)
*

Thus, in our terminology, if & is the set of circuits of a

doi:10.1016/0095-8956(88)90035-4
fatcat:u546twqtu5cgrm6igegzpptnk4
*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*. ...##
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Up to a double cover, every regular connected graph is isomorphic to a Schreier graph
[article]

2022
*
arXiv
*
pre-print

We prove that every connected locally finite regular

arXiv:2010.06431v2
fatcat:dya33jxqlffmrb7gbzbenymhgy
*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 . ...##
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On Perfect Matchings in Matching Covered Graphs
[article]

2017
*
arXiv
*
pre-print

In this paper, we show that, for every integer k> 3, there exist

arXiv:1703.05412v2
fatcat:yunkqjonozbszm37ttxsuunzt4
*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*. ...##
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Strong LP duality in weighted infinite bipartite graphs

1994
*
Discrete Mathematics
*

We prove a weighted generalization of Kiinig's duality theorem for

doi:10.1016/0012-365x(94)90367-0
fatcat:hkc4osoov5c5nh4fonstysahpe
*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(&). ...##
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A generalization of Tutte's 1-factor theorem to countable graphs

1984
*
Journal of combinatorial theory. Series B (Print)
*

A criterion is proved for a countable

doi:10.1016/0095-8956(84)90052-2
fatcat:5nqwfzj5qfgirheagetltt4abi
*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). ...##
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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. ...##
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Matchings in infinite graphs

1988
*
Journal of combinatorial theory. Series B (Print)
*

A general criterion is proved for a

doi:10.1016/0095-8956(88)90098-6
fatcat:u6k7jpm6y5futdpeaze27eljyq
*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*. ...##
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Arithmetically maximal independent sets in infinite graphs

2005
*
Discussiones Mathematicae Graph Theory
*

A positive answer is given to the following classes of

doi:10.7151/dmgt.1270
fatcat:xbzyjjmklrbedcvegorn24q63a
*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. ...##
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Conditions for matchability in groups and field extensions
[article]

2022
*
arXiv
*
pre-print

Note that the method of associating a bipartite

arXiv:2107.09029v3
fatcat:qi5gmi6a5jehbn5av5r3kwfegu
*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. ...##
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Strongly maximal matchings in infinite weighted graphs
[article]

2009
*
arXiv
*
pre-print

Given an assignment of weights w to the edges of a

arXiv:0911.4010v1
fatcat:yexxxciyvvcx5kpxxxs5lhzluy
*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). ...##
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Strongly Maximal Matchings in Infinite Graphs

2008
*
Electronic Journal of Combinatorics
*

Given an assignment of weights $w$ to the edges of an

doi:10.37236/860
fatcat:pn4rl3ut6nbv3ld5xn6scsuppe
*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). ...
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