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One or Two Disjoint Circuits Cover Independent Edges

2002
*
Journal of combinatorial theory. Series B (Print)
*

If k is even

doi:10.1006/jctb.2001.2059
fatcat:3vhfdtafs5cmroh5cblnu43upq
*or*G − L is connected, then there exist*one**or**two**disjoint**circuits*containing all the*edges*in L. This theorem is the first step in the proof of the*conjecture*of L. ...*Lovász*(1974, Period. Math. Hungar., 82) and D. R.*Woodall*(1977, J. Combin. Theory Ser. B 22, 274-278). ... Considering this situation,*Lovász*[14] and*Woodall*[18]*independently**conjectured*the following:*Conjecture*1. ...##
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Circuits through prescribed edges
[article]

2018
*
arXiv
*
pre-print

We prove that a connected graph contains a

arXiv:1810.09323v1
fatcat:kvjlxko35fc7bnsc4khedzl25q
*circuit*---a closed walk that repeats no*edges*---through any k prescribed*edges*if and only if it contains no odd cut of size at most k. ... The strongest of these is the following:*Conjecture*1.1 (*Lovász*-*Woodall**Conjecture*) . Let S be a set of k*independent**edges*in a k-connected graph G. ... the following variants of the*Lovász*-*Woodall**Conjecture*. ...##
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Page 8808 of Mathematical Reviews Vol. , Issue 2002M
[page]

2002
*
Mathematical Reviews
*

*two*

*disjoint*

*circuits*

*cover*

*independent*

*edges*. ... in the

*conjecture*, L is either contained in a cycle

*or*contained in

*two*

*disjoint*cycles. ...

##
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Fractional Matchings, Component-Factors and Edge-Chromatic Critical Graphs

2021
*
Graphs and Combinatorics
*

Consequences of these results for Vizing's critical graph

doi:10.1007/s00373-020-02266-6
fatcat:qd34qe64tvbwdi24uc7uottqy4
*conjectures*are discussed. ... Furthermore, it shows that for every*edge*e of G, there is a $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$ { K 1 , 1 , K 1 , 2 , C n : n ≥ 3 } -factor F with $$e \in E(F)$$ e ∈ E ( F ) . ... A set M (M EðGÞ*or*M & VðGÞ) is*independent*, if no*two*elements of M are adjacent. An*independent*set of*edges*is also called a matching of G. ...##
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Page 3961 of Mathematical Reviews Vol. , Issue 88h
[page]

1988
*
Mathematical Reviews
*

Let

*two*vertex-*disjoint*chain-chords be given, let*one*of them connect vertices x;, y2, and let the the other connect vertices X2, y2. ... In some special cases (such as when all*edges*are loops*or*if the degrees are severely restricted) an exact formula results. ...##
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Page 24 of Mathematical Reviews Vol. 55, Issue 1
[page]

1978
*
Mathematical Reviews
*

The maximum number of

*disjoint*directed cuts equals the minimum number of*edges*which*cover*all directed cuts. ... Mel'nikov (Novosibirsk)*Lovasz*, Laszlé 174*On**two*minimax theorems in graph. J. Combinatorial Theory Ser. B 21 (1976), no. 2, 96-103. ...##
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Page 1896 of Mathematical Reviews Vol. 52, Issue 6
[page]

1976
*
Mathematical Reviews
*

maximum number of

*disjoint**edges*and the*cover*problem is to find the minimum number of points to represent its*edges*. ... Let a matrix N be given with linearly*independent*rows the elements of which are 0*or*1. The problem of describing the class of all graphs for which N is the*circuit*-matrix is considered. ...##
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Packing and Covering in Combinatorics

1980
*
Journal of the Operational Research Society
*

We have tried to

doi:10.2307/2581088
fatcat:szdtpgrkzrfvvehhovqz3r6vxq
*cover*both the more*or*less classical theory as well as the more recent results. ... bound, Wilson's existence theory for designs, Lovasz's results*on*perfect graphs, Kneser's*conjecture*and the Shannon capacity, Baranyai's theorem*on*partitions into partitions, the Cook-Karp theory of ... of odd*circuits*of K 5 ,*or*the minimal*edge*-sets in K 5 intersecting each odd*circuit*. ...##
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Graph colorings with local constraints -- a survey

1997
*
Discussiones Mathematicae Graph Theory
*

restrictions are imposed

doi:10.7151/dmgt.1049
fatcat:du2wuuygubedxp5jzztrmw5ike
*on*the color assignment. ... subset C(v) ⊆ L(v) of cardinality c v for each vertex in such a way that the sets C(v), C(v ) are*disjoint*for each pair v, v of adjacent vertices. ... Add a new vertex*or*a new*edge*. 2. Let G 1 , G 2 be vertex-*disjoint*, and x i y i an*edge*in G i , i = 1, 2. ...##
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Some Recent Progress and Applications in Graph Minor Theory

2007
*
Graphs and Combinatorics
*

Some of these along with some other recent advances

doi:10.1007/s00373-006-0684-x
fatcat:wkf3w6cemzc4hnnw3neqpck6hu
*on*graph minors are surveyed. ... This result was used to prove Wagner's*Conjecture*that finite graphs are well-quasi-ordered under the graph minor relation. ... K 3k−1 minus k*independent**edges*shows that the condition "sufficiently large" is necessary. ...##
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Combinatorial optimization: packing and covering

2002
*
ChoiceReviews
*

In a clutter, a matching is a set of pairwise

doi:10.5860/choice.39-4018
fatcat:rntd6v7wc5bwjneouupuhg4sr4
*disjoint**edges*. A transversal is a set of vertices that intersects all the*edges*. ... The elements of V are the vertices of C and those of E are the*edges*. For example, a simple graph (V, E) (no multiple*edges**or*loops) is a clutter. ... These notes are based*on*a series of ten lectures delivered at a regional ...##
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Thickness and Antithickness of Graphs
[article]

2018
*
arXiv
*
pre-print

This paper explores the relationship between the thickness and antithickness of a graph, under various graph drawing models, with an emphasis

arXiv:1708.04773v2
fatcat:7ktieooxb5glxkodmzmfgolqiu
*on*extremal questions. ... (Here*edges*with a common endvertex v are considered to intersect at v.) ... Acknowledgement This research was initiated at the 2006 Bellairs Workshop*on*Computational Geometry organised by Godfried Toussaint. ...##
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Bipartite bihypergraphs: A survey and new results

2006
*
Discrete Mathematics
*

., a hyperedge of H 0

doi:10.1016/j.disc.2005.10.026
fatcat:aszv3zjsufa53lilud5yq2j3ci
*or*H 1 . Moreover, we can construct all bipartitions of a k-complete bihypergraph, if any, in polynomial time. ... We show that recognizing strongly bipartite bihypergraphs (H 0 , H 1 ) is a co-NP-complete problem even in the case where H 0 is a graph and H 1 has exactly*one*hyperedge. ... For example, Four Color*Conjecture*is equivalent to the following: the family of all odd*circuits*of a planar graph (considered as sets of*edges*) has property B. ...##
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The history of degenerate (bipartite) extremal graph problems
[article]

2013
*
arXiv
*
pre-print

This paper is a survey

arXiv:1306.5167v2
fatcat:t6puw44f4rayfehkdstmlorsfy
*on*Extremal Graph Theory, primarily focusing*on*the case when*one*of the excluded graphs is bipartite. ...*On**one*hand we give an introduction to this field and also describe many important results, methods, problems, and constructions. ... Given*two*graphs U and W , their product graph is the*one*obtained from vertex-*disjoint*copies of these*two*graphs by joining each vertex of U to each vertex of W . ...##
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Page 163 of Mathematical Reviews Vol. , Issue Subject Index
[page]

*
Mathematical Reviews
*

(with Vumar, Elkin)

*On*the circumference of 3-connected graphs. (English summary) 2002m:05128 Kawarabayashi, Ken-ichi*One**or**two**disjoint**circuits**cover**independent**edges*.*Lovasz*-*Woodall**conjecture*. ... (English summary) 2002d:05075 Ishigami, Yoshiyasu (with Wang, Hong’) An extension of a theorem*on*cycles containing specified*independent**edges*. ...
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