Bollobás and Thomason showed that a multigraph of order n and size at least n+ c (c ≥ 1) contains a cycle of length at most 2( n/c + 1) log2 2c . We show in this paper that a multigraph (with no loop) of order n and minimum degree at least 5 contains a chorded cycle (a cycle with a chord) of length at most 300 log2 n. As an application of this result, we show that a graph of sufficiently large ...

We study weak and strong survival for branching random walks on multigraphs. We prove that, for a large class of multigraphs, weak survival is related to a geometrical parameter of the multigraph and that the existence of a pure weak phase is equivalent to nonamenability. Finally we study weak and strong critical behaviors of the branching random walk.

We investigate 2-balanced colorings of sparse hypergraphs. As applications, we derive several results on balanced edge-colorings of multigraphs.

Given an Eulerian multigraph, a subset T of its vertices, and a collection H of subsets of T, we ask how few edge-disjoint paths can contain maximum (A, T"A)flows, for all A # H at once. We answer the question for a certain class of hypergraphs H by presenting a strongly polynomial construction of a minimum set of such paths and a min-max formula for its cardinality. The method consists in redu...

A maximal surface S with isolated singularities in a complete flat Lorentzian 3-manifold N is said to be entire if it lifts to a (periodic) entire multigraph S̃ in L. In addition, S is called of finite type if it has finite topology, finitely many singular points and S̃ is a finitely sheeted multigraph. Complete (or proper) maximal immersions with isolated singularities in N are entire, and entir...

We present some heuristics incorporating the philosophy of compatible relaxation into an existing algebraic multigrid method, the so-called multigraph solver of Bank and Smith [1]. In particular, approximate left and right eigenvectors of the iteration matrix for the smoother are used in computing both the sparsity pattern and the numerical values of the transfer matrices that correspond to res...

Let Γ(n, m) denote the class of all graphs and multigraphs with n nodes and m edges. A central question in network reliability theory is the network augmentation problem: For G ∈ Γ(n, m) fixed, what H ∈ Γ(n, m + k) such that G ⊂ H is t-optimal, that is, maximizes the tree number t(H)? In the network synthesis problem, where G is the empty graph on n vertices, it is conjectured that all t-optima...

Application of graph transformations for software verification and model transformation is an emergent field of research. In particular, graph transformation approaches provide a natural way of modelling object oriented systems and semantics of object-oriented languages. There exist a number of tools for graph transformations that are often specialised in a particular kind of graphs and/or grap...

Let G be a multigraph. We study the space X (G) of all pictures of G in complex projective d-space. The main result is that the homology groups (with integer coefficients) of X (G) are completely determined by the Tutte polynomial of G. One application is a criterion in terms of the Tutte polynomial for independence in the d-parallel matroids studied in combinatorial rigidity theory. In the cas...