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### One or Two Disjoint Circuits Cover Independent Edges

Ken-ichi Kawarabayashi
2002 Journal of combinatorial theory. Series B (Print)
If k is even 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.  ...

### Circuits through prescribed edges [article]

Paul Knappe, Max Pitz
2018 arXiv   pre-print
We prove that a connected graph contains a 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.  ...

### 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.  ...

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

Antje Klopp, Eckhard Steffen
2021 Graphs and Combinatorics
Consequences of these results for Vizing's critical graph 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.  ...

### 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.  ...

### 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.  ...

### 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.  ...

### Packing and Covering in Combinatorics

R. A. Cuninghame-Green, A. Schrijver
1980 Journal of the Operational Research Society
We have tried to 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.  ...

### Graph colorings with local constraints -- a survey

Zsolt Tuza
1997 Discussiones Mathematicae Graph Theory
restrictions are imposed 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.  ...

### Some Recent Progress and Applications in Graph Minor Theory

Ken-ichi Kawarabayashi, Bojan Mohar
2007 Graphs and Combinatorics
Some of these along with some other recent advances 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.  ...

### Combinatorial optimization: packing and covering

2002 ChoiceReviews
In a clutter, a matching is a set of pairwise 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  ...

### Thickness and Antithickness of Graphs [article]

Vida Dujmović, David R. Wood
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 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.  ...

### Bipartite bihypergraphs: A survey and new results

Inna Zverovich, Igor Zverovich
2006 Discrete Mathematics
., a hyperedge of H 0 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.  ...

### The history of degenerate (bipartite) extremal graph problems [article]

Zoltán Füredi, Miklós Simonovits
2013 arXiv   pre-print
This paper is a survey 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 .  ...

### 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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