This book provides a comprehensive introduction to the modern study of spanning trees. A spanning tree for a graph G is a subgraph of G that is a tree and contains all the vertices of G. There are many situations in which good spanning trees must be found. Whenever one wants to find a simple, cheap, yet efficient way to connect a set of terminals, be they computers, telephones, factories, or ci...

The Krushkal polynomial is the most general polynomial invariant of graphs on surfaces. It was introduced in [1]. A collection of edges forms a spanning quasi-tree if the regular neighborhood of the corresponding spanning subgraph has one boundary component. In the case of plane graphs it is just a spanning tree. The set of edges of a spanning quasi-trees is a feasible set of the corresponding ...

We consider three probability measures on subsets of edges of a given finite graph G, namely those which govern, respectively, a uniform forest, a uniform spanning tree, and a uniform connected subgraph. A conjecture concerning the negative association of two edges is reviewed for a uniform forest, and a related conjecture is posed for a uniform connected subgraph. The former conjecture is veri...

A parity subgraph of a graph is a spanning subgraph such that the degrees of all vertices have the same parity in both the subgraph and the original graph. Let G be a cyclically 6-edge-connected cubic graph. Steffen (Intersecting 1-factors and nowhere-zero 5-flows 1306.5645, 2013) proved that G has a nowhere-zero 5-flow if G has two perfect matchings with at most two intersections. In this pape...

In this article, Temperley’s bijection between spanning trees of a subgraph of the square grid on the one hand, and perfect matchings (also known as dimer coverings) of an associated subgraph of the square grid on the other, is extended to the setting of general planar directed (and undirected) graphs, where edges carry nonnegative weights that induce a weighting on the set of spanning trees. W...

An H-factor of a graph G is a spanning subgraph of G whose connected components are isomorphic to H. Given a properly edge-colored graph G, a rainbow H-subgraph of G is an H-subgraph of G whose edges have distinct colors. A rainbow H-factor is an H-factor whose components are rainbow H-subgraphs. The following result is proved. If H is any fixed graph with h vertices then every properly edge-co...

Summary In this article sets of certain subgraphs a graph are formalized in the Mizar system [7], [1], based on formalization graphs [11] briefly sketched [12]. The main result is spanning subgraph theorem.

We prove several results about chordal graphs and weighted by focusing on exposed edges. These are edges that properly contained in a single maximal complete subgraph. This leads to characterization of via deletions sequence from graph. Most interesting is this context the connected components edge-induced subgraph 2 -edge connected. use latter fact case give modified version Kruskal's second a...

We propose an O(min{m+n logn,m log∗n}) to find a minmax strongly connected spanning subgraph of a digraph with n nodes and m arcs. A generalization of this problem called the minmax strongly connected subgraph problem with node penalties is also considered. An O(m logn) algorithm is proposed to solve this general problem. We also discuss ways to improve the average complexity of this algorithm.