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Liftings in finite graphs and linkages in infinite graphs with prescribed edge-connectivity. (English) Zbl 1353.05074
Summary: 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.
##### MSC:
 05C40 Connectivity
##### Keywords:
edge-connectivity; lifting
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##### References:
  Aharoni, R; Thomassen, C, Infinite, highly connected digraphs with no two arc-disjoint spanning trees, J. Graph Theory, 13, 71-74, (1989) · Zbl 0665.05023  Chan, YH; Fung, WS; Lau, LC; Yung, CK, Degree bounded network design with metric costs, SIAM J. Comput., 40, 953-980, (2011) · Zbl 1235.05078  Frank, A, On a theorem of mader, Ann. Disc. Math., 101, 49-57, (1992) · Zbl 0789.05051  Huck, A, A sufficient condition for graphs to be weakly $$k$$-linked, Graphs Comb., 7, 323-351, (1991) · Zbl 0780.05036  Mader, W, A reduction method for edge-connectivity in graphs, Ann. Disc. Math., 3, 145-164, (1978) · Zbl 0389.05042  Okamura, H, Every $$4k$$-edge-connected graph is weakly-$$3k$$-linked, Graphs Comb., 6, 179185, (1990) · Zbl 0708.05036  Thomassen, C, 2-linked graphs, Eur. J. Comb., 1, 371-378, (1980) · Zbl 0457.05044  Thomassen C.: Orientations of infinite graphs with prescribed edge-connectivity. Combinatorica (2016). doi:10.1007/s00493-015-3173-0 · Zbl 1399.05139
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