Enumerating spanning trees of graphs with an involution

As the extension of the previous work by Ciucu and the present authors [M. Ciucu, W.G. Yan, F.J. Zhang, The number of spanning trees of plane graphs with reflective symmetry, J. Combin. Theory Ser. A 112 (2005) 105-116], this paper considers the problem of enumeration of spanning trees of weighted graphs with an involution which allows fixed points. We show that if G is a weighted graph with an involution, then the sum of weights of spanning trees of G can be expressed in terms of the product of the sums of weights of spanning trees of two weighted graphs with a smaller size determined by the involution of G. As applications, we enumerate spanning trees of the almost-complete bipartite graph, the almost-complete graph, the Möbius ladder, and the almost-join of two copies of a graph.     

The number of spanning trees of plane graphs with reflective symmetry

A plane graph is called symmetric if it is invariant under the reflection across some straight line (called symmetry axis). Let G be a symmetric plane graph. We prove that if there is no edge in G intersected by its symmetry axis then the number of spanning trees of G can be expressed in terms of the product of the number of spanning trees of two smaller graphs, each of which has about half the number of vertices of G.     

Matchings and spanning trees in Boolean weighted graphs

It is shown that in a 0-sum Boolean weighted graph G the sum of the weights taken over all the spanning trees equals the sum of the weights taken over all the perfect matchings in the graph G − v, where v is any vertex of G. Several related theorems are proved which include parity results on perfect matchings and spanning trees in Eulerian graphs. The ideas on perfect matchings in 0-sum Boolean weighted graphs are generalized to matchings in any Boolean weighted graph.     

Interpolation theorem for the number of end-vertices of spanning trees

The following interpolation theorem is proved: If a graph G contains spanning trees having exactly m and n end-vertices, with m < n, then for every integer k, m < k < n, G contains a spanning tree having exactly k end-vertices. This settles a problem posed by Chartrand at the Fourth International Conference on Graph Theory and Applications held in Kalamazoo, 1980.     

Sandpile groups and spanning trees of directed line graphs

We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph G and its directed line graph LG. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at a fixed vertex. In the case when G is regular of degree k, we show that the sandpile group of G is isomorphic to the quotient of the sandpile group of LG by its k-torsion subgroup. As a corollary we compute the sandpile groups of two families of graphs widely studied in computer science, the de Bruijn graphs and Kautz graphs.     

Minimum edge ranking spanning trees of split graphs

Given a graph G, the minimum edge ranking spanning tree problem (MERST) is to find a spanning tree of G whose edge ranking is minimum. However, this problem is known to be NP-hard for general graphs. In this paper, we show that the problem MERST has a polynomial time algorithm for split graphs, which have useful applications in practice. The result is also significant in the sense that this is a first non-trivial graph class for which the problem MERST is found to be polynomially solvable. We also show that the problem MERST for threshold graphs can be solved in linear time, where threshold graphs are known to be split.     

Long cycles and spanning subgraphs of locally maximal 1-planar graphs

A graph is 1-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges induce a complete subgraph, then the graph is locally maximal 1-planar. For a 3-connected locally maximal 1-planar graph G, we show the existence of a spanning 3-connected planar subgraph and prove that G is Hamiltonian if G has at most three 3-vertex-cuts, and that G is traceable if G has at most four 3-vertex-cuts. Moreover, infinitely many nontraceable 5-connected 1-planar graphs are presented.     

A note on graphs with diameter-preserving spanning trees

The distance between a pair of vertices u, v in a graph G is the length of a shortest path joining u and v. The diameter diam(G) of G is the maximum distance between all pairs of vertices in G. A spanning tree T of G is diameter preserving if diam(T) = diam(G). In this note, we characterize graphs that have diameter-preserving spanning trees.     

Rainbow faces in edge‐colored plane graphs

A face of an edge‐colored plane graph is called rainbow if the number of colors used on its edges is equal to its size. The maximum number of colors used in an edge coloring of a connected plane graph Gwith no rainbow face is called the edge‐rainbowness of G. In this paper we prove that the edge‐rainbowness of Gequals the maximum number of edges of a connected bridge face factor H of G, where a bridge face factor H of a plane graph Gis a spanning subgraph H of Gin which every face is incident with a bridge and the interior of any one face f∈F(G) is a subset of the interior of some face f′∈F(H). We also show upper and lower bounds on the edge‐rainbowness of graphs based on edge connectivity, girth of the dual graphs, and other basic graph invariants. Moreover, we present infinite classes of graphs where these equalities are attained. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 84–99, 2009     

Locally connected spanning trees in strongly chordal graphs and proper circular-arc graphs

A locally connected spanning tree of a graph G is a spanning tree T of G such that the set of all neighbors of v in T induces a connected subgraph of G for every v∈V(G). The purpose of this paper is to give linear-time algorithms for finding locally connected spanning trees on strongly chordal graphs and proper circular-arc graphs, respectively.     

Universality for bounded degree spanning trees in randomly perturbed graphs

We solve a problem of Krivelevich, Kwan and Sudakov concerning the threshold for the containment of all bounded degree spanning trees in the model of randomly perturbed dense graphs. More precisely, we show that, if we start with a dense graph G_α on n vertices with δ(G_α) ≥ αn for α > 0 and we add to it the binomial random graph G(n,C/n), then with high probability the graph G_α∪G(n,C/n) contains copies of all spanning trees with maximum degree at most Δ simultaneously, where C depends only on α and Δ.     

On encodings of spanning trees

Deo and Micikevicius recently gave a new bijection for spanning trees of complete bipartite graphs. In this paper we devise a generalization of Deo and Micikevicius's method, which is also a modification of Olah's method for encoding the spanning trees of any complete multipartite graph K(n₁,…,n_r). We also give a bijection between the spanning trees of a planar graph and those of any of its planar duals. Finally we discuss the possibility of bijections for spanning trees of DeBriujn graphs, cubes, and regular graphs such as the Petersen graph that have integer eigenvalues.     

Some results on spanning trees

Some structures of spanning trees with many or less leaves in a connected graph are determined.We show（1） a connected graph G has a spanning tree T with minimum leaves such that T contains a longest path,and（2） a connected graph G on n vertices contains a spanning tree T with the maximum leaves such that Δ（G） =Δ（T） and the number of leaves of T is not greater than n D（G）＋1,where D（G） is the diameter of G.     

On spanning tree congestion of graphs

Let G be a connected graph and T be a spanning tree of G. For e∈E(T), the congestion of e is the number of edges in G connecting two components of T−e. The edge congestion ofGinT is the maximum congestion over all edges in T. The spanning tree congestion ofG is the minimum congestion of G in its spanning trees. In this paper, we show the spanning tree congestion for the complete k-partite graphs and the two-dimensional tori. We also address lower bounds of spanning tree congestion for the multi-dimensional grids and the hypercubes.     

On the probabilistic min spanning tree Problem

We study a probabilistic optimization model for min spanning tree, where any vertex v _i of the input-graph G(V, E) has some presence probability p _i in the final instance G′ ⊂ G that will effectively be optimized. Suppose that when this “real” instance G′ becomes known, a spanning tree T, called anticipatory or a priori spanning tree, has already been computed in G and one can run a quick algorithm (quicker than one that recomputes from scratch), called modification strategy, that modifies the anticipatory tree T in order to fit G′. The goal is to compute an anticipatory spanning tree of G such that, its modification for any G￠ í GG' \subseteq G is optimal for G′. This is what we call probabilistic min spanning tree problem. In this paper we study complexity and approximation of probabilistic min spanning tree in complete graphs under two distinct modification strategies leading to different complexity results for the problem. For the first of the strategies developed, we also study two natural subproblems of probabilistic min spanning tree, namely, the probabilistic metric min spanning tree and the probabilistic min spanning tree 1,2 that deal with metric complete graphs and complete graphs with edge-weights either 1, or 2, respectively.     

A lower bound on the number of spanning trees with k end-vertices

If a graph G with cycle rank ρ contains both spanning trees with m and with n end-vertices, m < n, then G has at least 2ρ spanning trees with k end-vertices for each integer k, m < k < n. Moreover, the lower bound of 2ρ is best possible.     

Spanning trees of extended graphs

A generalization of the Prüfer coding of trees is given providing a natural correspondence between the set of codes of spanning trees of a graph and the set of codes of spanning trees of theextension of the graph. This correspondence prompts us to introduce and to investigate a notion ofthe spanning tree volume of a graph and provides a simple relation between the volumes of a graph and its extension (and in particular a simple relation between the spanning tree numbers of a graph and its uniform extension). These results can be used to obtain simple purely combinatorial proofs of many previous results obtained by the Matrix-tree theorem on the number of spanning trees of a graph. The results also make it possible to construct graphs with the maximal number of spanning trees in some classes of graphs.     

On the complexity of rainbow spanning forest problem

Given a graph \(G=(V,E,L)\) and a coloring function \(\ell : E \rightarrow L\), that assigns a color to each edge of G from a finite color set L, the rainbow spanning forest problem (RSFP) consists of finding a rainbow spanning forest of G such that the number of components is minimum. A spanning forest is rainbow if all its components (trees) are rainbow. A component whose edges have all different colors is called rainbow component. The RSFP on general graphs is known to be NP-complete. In this paper we use the 3-SAT Problem to prove that the RSFP is NP-complete on trees and we prove that the problem is solvable in polynomial time on paths, cycles and if the optimal solution value is equal to 1. Moreover, we provide an approximation algorithm for the RSFP on trees and we show that it approximates the optimal solution within 2.     

Saturated graphs with minimal number of edges

Let F = {F₁,…} be a given class of forbidden graphs. A graph G is called F-saturated if no F_i ∈ F is a subgraph of G but the addition of an arbitrary new edge gives a forbidden subgraph. In this paper the minimal number of edges in F-saturated graphs is examined. General estimations are given and the structure of minimal graphs is described for some special forbidden graphs (stars, paths, m pairwise disjoint edges).