Graphs and Combinatorics

, Volume 32, Issue 6, pp 2575–2589 | Cite as

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

  • Seongmin Ok
  • R. Bruce Richter
  • Carsten Thomassen
Original Paper

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 

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Copyright information

© Springer Japan 2016

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkCopenhagenDenmark
  2. 2.Department of Combinatorics & OptimizationUniversity of WaterlooWaterlooCanada

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