# The edge-disjoint paths problem in Eulerian graphs and 4-edge-connected graphs

## Abstract

*k*pairs of vertices, and we have to decide whether or not the graph has

*k*edge-disjoint paths connecting given pairs of terminals. Robertson and Seymour’s graph minor project gives rise to a polynomial time algorithm for this problem for any fixed

*k*, but their proof of the correctness needs the whole Graph Minor project. We give a faster algorithm and a much simpler proof of the correctness for the edge-disjoint paths problem. Our results can be summarized as follows:

- 1.
If an input graph is either 4-edge-connected or Eulerian, then our algorithm only needs to look for the following three simple reductions: (i) Excluding vertices of high degree. (ii) Excluding ≤3-edge-cuts. (iii) Excluding large clique minors.

- 2.
When an input graph is either 4-edge-connected or Eulerian, the number of terminals

*k*is allowed to be a non-trivially super constant number, up to*k*=*O*((log log log*n*)^{½−ε}) for any*ε*> 0. In addition, if an input graph is either 4-edge-connected planar or Eulerian planar,*k*is allowed to be*O*((log*n*^{½−ε}) for any*ε*> 0. - 3.
We also give our own algorithm for the edge-disjoint paths problem in general graphs. We basically follow the Robertson-Seymour’s algorithm, but we cut half of the proof of the correctness for their algorithm. In addition, our algorithm is faster than Robertson and Seymour’s.

### Mathematics Subject Classification (2000)

05C38 05C83 05C85## Preview

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