Friedrich Regen

Independence, odd girth, and average degree

We prove several tight lower bounds in terms of the order and the average degree for the independence number of graphs that are connected and/or satisfy some odd girth condition. Our main result is the extension of a lower bound for the independence number of triangle-free graphs of maximum degree at most three due to Heckman and Thomas [Discrete Math 233 (2001), 233–237] to arbitrary triangle-free graphs. For connected triangle-free graphs of order n and size m, our result implies the existence of an independent set of order at least (4n−m−1)/7.

On spanning tree congestion

We prove that every connected graph G of order n has a spanning tree T such that for every edge e of T the edge cut defined in G by the vertex sets of the two components of T-e contains at most n^3^2 edges. This result solves a problem posed by Ostrovskii (M.I. Ostrovskii, Minimal congestion trees, Discrete Math. 285 (2004) 219-226).

On packing shortest cycles in graphs

We study the problems to find a maximum packing of shortest edge-disjoint cycles in a graph of given girth g (g-ESCP) and its vertex-disjoint analogue g-VSCP. In the case g=3, Caprara and Rizzi (2001) have shown that g-ESCP can be solved in polynomial time for graphs with maximum degree 4, but is APX-hard for graphs with maximum degree 5, while g-VSCP can be solved in polynomial time for graphs with maximum degree 3, but is APX-hard for graphs with maximum degree 4. For g@?{4,5}, we show that both problems allow polynomial time algorithms for instances with maximum degree 3, but are APX-hard for instances with maximum degree 4. For each g>=6, both problems are APX-hard already for graphs with maximum degree 3.

Cycle spectra of Hamiltonian graphs

We prove that every Hamiltonian graph with n vertices and m edges has cycles with more than p-12lnp-1 different lengths, where p=m-n. For general m and n, there exist such graphs having at most 2@?p+1@? different cycle lengths.

On the Cycle Spectrum of Cubic Hamiltonian Graphs

We prove lower bounds on the number of different cycle lengths of cubic Hamiltonian graphs that do not contain a fixed subdivision of a claw as an induced subgraph.

Packing edge-disjoint cycles in graphs and the cyclomatic number

For a graph G let @m(G) denote the cyclomatic number and let @n(G) denote the maximum number of edge-disjoint cycles of G. We prove that for every k>=0 there is a finite set P(k) such that every 2-connected graph G for which @m(G)-@n(G)=k arises by applying a simple extension rule to a graph in P(k). Furthermore, we determine P(k) for k@?2 exactly.

Lower bounds on the independence number of certain graphs of odd girth at least seven

Heckman and Thomas [C.C. Heckman, R. Thomas, A new proof of the independence ratio of triangle-free cubic graphs, Discrete Math. 233 (2001) 233-237] proved that every connected subcubic triangle-free graph G has an independent set of order at least (4n(G)-m(G)-1)/7 where n(G) and m(G) denote the order and size of G, respectively. We conjecture that every connected subcubic graph G of odd girth at least seven has an independent set of order at least (5n(G)-m(G)-1)/9 and verify our conjecture under some additional technical assumptions.

On Hamiltonian paths in distance graphs

Abstract For a finite set D ⊆ N with gcd ( D ) = 1 , we prove the existence of some n ∈ N such that the distance graph P n D with vertex set { 0 , 1 , … , n − 1 } in which two vertices u and v are adjacent exactly if | u − v | ∈ D , has a Hamiltonian path with endvertices 0 and n − 1 . This settles a conjecture posed by Penso et al. [L.D. Penso, D. Rautenbach, J.L. Szwarcfiter, Long cycles and paths in distance graphs, Discrete Math. 310 (2010) 3417–3420].

Edge-Injective and Edge-Surjective Vertex Labellings

For a graph

Graphs with many vertex-disjoint cycles

G=(V,E)