Using the technique of evolving sets, we explore connection between entropy growth and transience for simple random walks on connected infinite graphs with bounded degree. In particular show that a walk starting at vertex x0, if after n steps, En is least Cn where C independent then transient. We also give an example which demonstrates condition being x0 necessary.

Let {Gn : n ≥ 1} be a sequence of simple graphs. Suppose Gn has mn edges and each vertex of Gn is colored independently and uniformly at random with cn colors. Recently, Bhattacharya, Diaconis and Mukherjee (2014) proved universal limit theorems for the number of monochromatic edges in Gn. Their proof was by the method of moments, and therefore was not able to produce rates of convergence. By a...

We describe and analyse a simple greedy algorithm 2greedy that finds a good 2-matching M in the random graph G = G n,cn when c ≥ 10. A 2-matching is a spanning subgraph of maximum degree two and G is drawn uniformly from graphs with vertex set [n], cn edges and minimum degree at least three. By good we mean that M has O(log n) components. We then use this 2-matching to build a Hamilton cycle in...

Let G be an edge-colored graph. A heterochromatic (rainbow, or multicolored) path of G is such a path in which no two edges have the same color. Let d(v) denote the color degree and CN(v) denote the color neighborhood of a vertex v of G. In a previous paper, we showed that if d(v) ≥ k (color degree condition) for every vertex v of G, then G has a heterochromatic path of length at least ⌈ 2 ⌉, a...

The communication problem is to select a minimal set of placed sensor devices in a service area so that the entire service area is accessible by the minimal set of sensors. Finding the minimal set of sensors is modeled as a vertex-cover problem, where the vertex-cover set facilitates the communications between the sensors in a multi-hop fashion keeping in mind the limited communication range an...

In this paper we present an algorithm to generate all minimal 3-vertex connected spanning subgraphs of an undirected graph with n vertices and m edges in incremental polynomial time, i.e., for every K we can generate K (or all) minimal 3-vertex connected spanning subgraphs of a given graph in O(K2log(K)m2 +K2m3) time, where n and m are the number of vertices and edges of the input graph, respec...

Connected Vertex Cover is one of the classical problems of computer science, already mentioned in the monograph of Garey and Johnson [W.H. Freeman & Co., 1979]. Although the optimization and decision variants of finding connected vertex covers of minimum size or weight are well-studied, surprisingly there is no work on the enumeration or maximum number of minimal connected vertex covers of a gr...

The new general upper bound μ ≤ [ c 24 ] + 1 for the minimal weight μ of a selfdual vertex operator superalgebra of central charge c 6= 23 1 2 is proven. For central charges c ≤ 48, further improved estimates are given and examples of vertex operator superalgebras with large minimal weight are discussed. We also study the case of vertex operator superalgebras with N=1 supersymmetry which was fi...

A feedback vertex set in a graph is a set of vertices whose removal leaves the remaining graph acyclic. Given the vast number of published results concerning feedback vertex sets, it is surprising that the related combinatorics appears to be so poorly understood. The maximum number of minimal feedback vertex sets in a graph on n vertices is known to be at most 1.864. However, no examples of gra...