One or Two Disjoint Circuits Cover Independent Edges: Lovász-Woodall Conjecture

@article{Kawarabayashi2002OneOT,
  title={One or Two Disjoint Circuits Cover Independent Edges: Lov{\'a}sz-Woodall Conjecture},
  author={Ken-ichi Kawarabayashi},
  journal={J. Comb. Theory, Ser. B},
  year={2002},
  volume={84},
  pages={1-44}
}
In this paper, we prove the following theorem: Let L be a set of k independent edges in a k-connected graph G. 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. Lovasz (1974, Period. Math. Hungar., 82) and D. R. Woodall (1977, J. Combin. Theory Ser. B22, 274?278). In addition, we give the outline of the proof of the conjecture and refer to the forthcoming papers. 

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References

SHOWING 1-10 OF 20 REFERENCES

Any four independent edges of a 4-connected graph are contained in a circuit

Conjecture. Suppose G is a k-connected graph (k=>2), el, e~ . . . . , ekEE(G) are independent edges, and if k is odd then G {el, e2, ..., ek} is connected. Then G contains a circuit using all the

Circuits through specified edges

Cycles through specified vertices of a graph

TLDR
It is proved that ifS is a set ofk−1 vertices in ak-connected graphG, then the cycles throughS generate the cycle space ofG, and the existence of odd and even cycles through specified vertices is established.

Cycles intersecting a prescribed vertex set

TLDR
It is shown that for r = 1 or 2 every n-connected graph satisfies P(n + r,n), and if n ⩾ max{3,(2r −1)(r + 1)}, then every n -connectedgraph satisfies P (n +r,n).

On circuits through five edges

The binding number of a graph and its Anderson number

Cycles Through Prescribed and Forbidden Point Sets

Note on circuits containing specified edges

Circuits containing specified edges