# Liftings in Finite Graphs and Linkages in Infinite Graphs with Prescribed Edge-Connectivity

## Abstract

Let *G* be a graph and let *s* be a vertex of *G*. We consider the structure of the set of all lifts of two edges incident with *s* that preserve edge-connectivity. Mader proved that two mild hypotheses imply there is at least one pair that lifts, while Frank showed (with the same hypotheses) that there are at least \((\deg (s)-1)/2\) disjoint pairs that lift. We consider the *lifting graph*: its vertices are the edges incident with *s*, two being adjacent if they form a liftable pair. We have three main results, the first two with the same hypotheses as for Mader’s Theorem. (i) Let *F* be a subset of the edges incident with *s*. We show that *F* is independent in the lifting graph of *G* if and only if there is a single edge-cut *C* in *G* of size at most \(r+1\) containing all the edges in *F*, where *r* is the maximum number of edge-disjoint paths from a vertex (not *s*) in one component of \(G-C\) to a vertex (not *s*) in another component of \(G-C\). (ii) In the *k*-lifting graph, two edges incident with *s* are adjacent if their lifting leaves the resulting graph with the property that any two vertices different from *s* are joined by *k* pairwise edge-disjoint paths. If both \(\deg (s)\) and *k* are even, then the *k*-lifting graph is a connected complete multipartite graph. In all other cases, there are at most two components. If there are exactly two components, then each component is a complete multipartite graph. If \(\deg (s)\) is odd and there are two components, then one component is a single vertex. (iii) Huck proved that if *k* is odd and *G* is \((k+1)\)-edge-connected, then *G* is weakly *k*-linked (that is, for any *k* pairs \(\{x_i,y_i\}\), there are *k* edge-disjoint paths \(P_i\), with \(P_i\) joining \(x_i\) and \(y_i\)). We use our results to extend a slight weakening of Huck’s theorem to some infinite graphs: if *k* is odd, every \((k+2)\)-edge-connected, locally finite, 1-ended, infinite graph is weakly *k*-linked.

### Keywords

Edge-connectivity Lifting### Mathematics Subject Classification

05C40### References

- 1.Aharoni, R., Thomassen, C.: Infinite, highly connected digraphs with no two arc-disjoint spanning trees. J. Graph Theory
**13**, 71–74 (1989)MathSciNetCrossRefMATHGoogle Scholar - 2.Chan, Y.H., Fung, W.S., Lau, L.C., Yung, C.K.: Degree bounded network design with metric costs. SIAM J. Comput.
**40**(4), 953–980 (2011)MathSciNetCrossRefMATHGoogle Scholar - 3.Frank, A.: On a theorem of Mader. Ann. Disc. Math.
**101**, 49–57 (1992)MathSciNetCrossRefMATHGoogle Scholar - 4.Huck, A.: A sufficient condition for graphs to be weakly \(k\)-linked. Graphs Comb.
**7**, 323–351 (1991)MathSciNetCrossRefMATHGoogle Scholar - 5.Mader, W.: A reduction method for edge-connectivity in graphs. Ann. Disc. Math.
**3**, 145–164 (1978)MathSciNetCrossRefMATHGoogle Scholar - 6.Okamura, H.: Every \(4k\)-edge-connected graph is weakly-\(3k\)-linked. Graphs Comb.
**6**, 179185 (1990)MathSciNetCrossRefGoogle Scholar - 7.Thomassen, C.: 2-linked graphs. Eur. J. Comb.
**1**, 371–378 (1980)MathSciNetCrossRefMATHGoogle Scholar - 8.Thomassen C.: Orientations of infinite graphs with prescribed edge-connectivity. Combinatorica (2016). doi:10.1007/s00493-015-3173-0