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Counting 1-factors in infinite graphs
1990
Journal of combinatorial theory. Series B (Print)
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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
1990
Journal of combinatorial theory. Series B (Print)
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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
1991
Discrete Mathematics
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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
1986
Journal of combinatorial theory. Series A
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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
1988
Journal of combinatorial theory. Series B (Print)
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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]
2022
arXiv
pre-print
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2020 pre-print arXiv:2010.06431v1 fatcat:ovluy5bnvzhoraccvt5ufv5z2e
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]
2017
arXiv
pre-print
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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
1994
Discrete Mathematics
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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
1984
Journal of combinatorial theory. Series B (Print)
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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
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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
1988
Journal of combinatorial theory. Series B (Print)
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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
2005
Discussiones Mathematicae Graph Theory
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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]
2022
arXiv
pre-print
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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]
2009
arXiv
pre-print
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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
2008
Electronic Journal of Combinatorics
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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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