# 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

- Mathematics
- 1985

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…

### Cycles through specified vertices of a graph

- MathematicsComb.
- 1981

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 through a prescribed vertex set in N-connected graphs

- MathematicsJ. Comb. Theory, Ser. B
- 2004

### Cycles intersecting a prescribed vertex set

- MathematicsJ. Graph Theory
- 1991

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).