We prove that, for each positive integer k, every sufficiently large 3-connected regular matroid has a parallel minor isomorphic to M∗(K3,k), M(Wk), M(Kk), the cycle matroid of the graph obtained from K2,k by adding paths through the vertices of each vertex class, or the cycle matroid of the graph obtained from K3,k by adding a complete graph on the vertex class with three vertices.

0

We prove that, for each positive integer k, every sufficiently large 3-connected regular matroid has a parallel minor isomorphic to M∗(K3,k), M(Wk), M(Kk), the cycle matroid of the graph obtained from K2,k by adding paths through the vertices of each vertex class, or the cycle matroid of the graph obtained from K3,k by adding a complete graph on the vertex class with three vertices.

A seminal result of Reed et al. [15] in 1996 states that the Erdős-Pósa property holds for directed cycles, i.e. for every integer n there is an integer t such that every directed graph G has n pairwise vertex disjoint directed cycles or contains a set T ⊆ V (G) of at most t vertices such that G−T contains no directed cycle. In this paper, we consider the Erdős-Pósa property for directed cycles...

The problem of partitioning an assembly of polyhedral objects into two subassemblies that can be separated arises in assembly planning. We describe an algorithm to compute the set of all translations separating two polyhedra with n vertices in O(n*) steps and show that this is optimal. Given an assembly of k polyhedra with a total of n vertices, an extension of this algorithm identifies a valid...

The vertex set and arc set of a digraph D are denoted by V (D) and E (D), respectively, and the number of vertices in a digraph D is denoted by n (D). A directed cycle (path, walk) in a digraph will simply be called a cycle (path, walk). A graph or digraph is called hamiltonian if it contains a cycle that visits every vertex, traceable if it contains a path that visits every vertex, and walkabl...

For any h in N , a graph G = (V, E) is said to be h-magic if there exists a labeling l: E(G) to Z_{h}-{0} such that the induced vertex set labeling l^{+: V(G) to Z_{h}} defined by l^{+}(v)= Summation of l(uv)such that e=uvin in E(G) is a constant map. For a given graph G, the set of all for which G is h-magic is called the integer-magic spectrum of G and is denoted by IM(G). In this paper, the ...

The Caccetta-Häggkvist conjecture states that if G is a finite directed graph with at least n/k edges going out of each vertex, then G contains a directed cycle of length at most k. Hamidoune used methods and results from additive number theory to prove the conjecture for Cayley graphs and for vertex-transitive graphs. This expository paper contains a survey of results on the Caccetta-Häggkvist...

Fixed-parameter tractability Definition 2. A parameterized problem Π is fixed-parameter tractable (FPT) if there is an algorithm solving Π in time f(k) · poly(n), where n is the instance size, k is the parameter, poly is a polynomial function, and f is a computable function. Theorem 3. Let Π be a decidable parameterized problem. Π has a kernelization ⇔ Π is FPT. 2 A kernel for Hamiltonian Cycle...