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On a theorem of Mader. (English) Zbl 0789.05051
For vertices $$x$$, $$y$$ in a multigraph $$G$$, $$\lambda(x,y;G)$$ denotes the maximal number of edge-disjoint $$x,y$$-paths in $$G$$. Let $$G$$ be a finite multigraph and let $$s$$ be a vertex in $$G$$ of degree $$d(s) \neq 0,3$$ not incident to a bridge. For edges $$e_ i$$ joining $$s$$ and $$x_ i$$ for $$i=1,2$$, $$G^{e_ 1e_ 2}$$ arises from $$G$$ by deleting $$e_ 1$$, $$e_ 2$$, and, if $$x_ 1 \neq x_ 2$$, by adding a new edge between $$x_ 1$$ and $$x_ 2$$. It was proved in [W. Mader, A reduction method for edge-connectivity in graphs, Ann. Discrete Math. 3, 145-164 (1978; Zbl 0389.05042)] that there are edges $$e_ 1 \neq e_ 2$$ incident to $$s$$ such that $$\lambda (x,y;G^{e_ 1e_ 2})=\lambda (x,y;G)$$ for all vertices $$x,y$$ from $$G-s$$. Such a pair of edges is called a splittable pair at $$s$$. Now a relatively simple proof of this result is given, and it is shown that for $$d(s) \neq 3$$ there are $$\lfloor {d(s) \over 2} \rfloor$$ disjoint splittable pairs at $$s$$, whereas from the above result one gets for odd $$d(s)$$ only $${d(s)-3 \over 2}$$ such pairs.