Contributed Session Thu.2.H 3004

Thursday, 13:15 - 14:45 h, Room: H 3004

Cluster 2: Combinatorial optimization [...]

Cycles in graphs


Chair: Peter Recht



Thursday, 13:15 - 13:40 h, Room: H 3004, Talk 1

Peter Recht
A "min-max-theorem'' for the cycle packing problem in Eulerian graphs

Coauthor: Eva-Maria Sprengel


This lecture deals with the problem to determine a set \  Z*= {C1, C2, … , C ν (G) } of edge-disjoint cycles of maximum cardinality ν (G) in a graph G=(V,E). The problem is tackled by considering special subgraphs: \ for a vertex v ∈ V, let T(v) be a local trace at v, i.e. T(v) is an Eulerian subgraph of G such that every walk W(v), with start vertex v can be extended to an Eulerian tour in T(v). \ In general, maximal local traces are not uniquely defined but their packing numbers ν (T(v)) are. \ We prove that if G is is Eulerian every maximum edge-disjoint cycle packing Z* of G induces maximum local traces T(v) at v for every v ∈ V. In the opposite, if the total size v ∈ V |E(T(v)| is minimal then the set of induced edge-disjoint cycles in G must be maximum. \ The determination of such a maximal trace leads to a multi-commodity flow-problem with quadratic objective function.



Thursday, 13:45 - 14:10 h, Room: H 3004, Talk 2

Eva-Maria Sprengel
An optimal cycle packing for generalized Petersen graphs P(n,k) with k even

Coauthor: Peter Recht


For an undirected graph G=(V,E) a maximum cycle packing is a collection of pairwise edge-disjoint cycles. The maximum cardinality of such a packing is denoted as the cycle packing number ν (G).
In general the determination of a maximum cycle packing and the cycle packing number, respectively, is NP-hard.
In this lecture we consider the family of generalized Petersen graphs P(n,k) with k even. We give a lower and an upper bound on the cycle packing number and outline the structure of one optimal cycle packing of such graphs.



Thursday, 14:15 - 14:40 h, Room: H 3004, Talk 3

Lamia Aoudia
4-cycle polytope on a graph

Coauthor: Meziane Aider


The aim of this work is to give a convex hull of 4-cycle on a wider class of complete bipartite graphs.\
Given a bipartite graph Kn,m, where |{V1}|=n and
|{V1}|=m, E=V1 × V2 and a weight function w: E → R, The minimum weighted 4-cycle problem consist on finding a 4-cycle C ⊂ E such hat e ∈ Cwe is minimum. This problem can easily be solved in polynomial time by complete enumeration of the the 4-cycles of G. For each 4-cycle C, let XC denote the incidence vector of C defined by: XC (e) = 1 and XC =0 if e\notin C. The 4-cycle polytope PC4 n , m is the convex hull of the incidence vectors of the 4-cycles of Kn,m, i.e. PC4 n , m = convex hull {XC ∈ {0,1}: C is a 4-cycle of G }
The minimum weighted 4-cycle problem is clearly equivalent to the linear program min {wx :  x ∈ PC4n,n }.
We are mainly interested by the facial tructure of PC4 n,m. Thus, we enumerate some inequalities defining facets of PC4 n, n.


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