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Exponentially many perfect matchings in cubic graphs, manuscript
"... Abstract. We show that every cubic bridgeless graph G has at least 2 V (G)/3656 perfect matchings. This confirms an old conjecture of Lovász and Plummer. 1. ..."
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Abstract. We show that every cubic bridgeless graph G has at least 2 V (G)/3656 perfect matchings. This confirms an old conjecture of Lovász and Plummer. 1.
Disjoint Tpaths in tough graphs
"... Let G be a graph and T a set of vertices. A Tpath in G is a path that begins and ends in T, and none of its internal vertices are contained in T. We define a Tpath covering to be a union of vertexdisjoint Tpaths spanning all of T. Concentrating on graphs that are tough (the removal of any nonemp ..."
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Let G be a graph and T a set of vertices. A Tpath in G is a path that begins and ends in T, and none of its internal vertices are contained in T. We define a Tpath covering to be a union of vertexdisjoint Tpaths spanning all of T. Concentrating on graphs that are tough (the removal of any nonempty set X of vertices yields at most X  components), we completely characterize the edges that are contained in some Tpath covering. Our main tool is Mader’s Spaths theorem. A corollary of our result is that each edge of a kregular kedgeconnected graph (k ≥ 2) is contained in a Tpath covering. This is, in a sense, a best possible counterpart of the result of Plesník that every edge of a kregular (k − 1)edgeconnected graph of even order is contained in a 1factor.
Spanning even subgraphs . . .
, 2006
"... By Petersen’s theorem, a bridgeless cubic graph has a 2factor. H. Fleischner extended this result to bridgeless graphs of minimum degree at least three by showing that every such graph has a spanning even subgraph. Our main result is that, under the stronger hypothesis of 3edgeconnectivity, we ca ..."
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By Petersen’s theorem, a bridgeless cubic graph has a 2factor. H. Fleischner extended this result to bridgeless graphs of minimum degree at least three by showing that every such graph has a spanning even subgraph. Our main result is that, under the stronger hypothesis of 3edgeconnectivity, we can find a spanning even subgraph in which every component has at least five vertices. We show that this is in some sense best possible by constructing an infinite family of 3edgeconnected graphs in which every spanning even subgraph has a 5cycle as a component.
A superlinear bound . . . matchings in cubic bridgeless graphs
, 2012
"... Lovász and Plummer conjectured in the 1970’s that cubic bridgeless graphs have exponentially many perfect matchings. This conjecture has been verified for bipartite graphs by Voorhoeve in 1979, and for planar graphs by Chudnovsky and Seymour in 2008, but in general only linear bounds are known. In t ..."
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Lovász and Plummer conjectured in the 1970’s that cubic bridgeless graphs have exponentially many perfect matchings. This conjecture has been verified for bipartite graphs by Voorhoeve in 1979, and for planar graphs by Chudnovsky and Seymour in 2008, but in general only linear bounds are known. In this paper, we provide the first superlinear bound in the general case.