On the spanning tree packing number of a graph: a survey

The spanning tree packing number or STP number of a graph G is the maximum number of edge-disjoint spanning trees contained in G. We use an observation of Paul Catlin to investigate the STP numbers of several families of graphs including quasi-random graphs, regular graphs, complete bipartite graphs, cartesian products and the hypercubes.     

Some results on R 2-edge-connectivity of even regular graphs

Let G be a connected k(≥3)-regular graph with girth g. A set S of the edges in G is called an Rredge-cut if G-S is disconnected and comains neither an isolated vertex nor a one-degree vertex. The R2-edge-connectivity of G, denoted by λ^n(G), is the minimum cardinality over all R2-edge-cuts, which is an important measure for fault-tolerance of computer interconnection networks. In this paper, λ^n(G)=g(2k-2) for any 2k-regular connected graph G (≠K5) that is either edge-transitive or vertex-transitive and g≥5 is given.     

Strong clique trees,neighborhood trees,and strongly chordal graphs

Maximal complete subgraphs and clique trees are basic to both the theory and applications of chordal graphs. A simple notion of strong clique tree extends this structure to strongly chordal graphs. Replacing maximal complete subgraphs with open or closed vertex neighborhoods discloses new relationships between chordal and strongly chordal graphs and the previously studied families of chordal bipartite graphs, clique graphs of chordal graphs (dually chordal graphs), and incidence graphs of biacyclic hypergraphs. © 2000 John Wiley & Sons, Inc. J. Graph Theory 33: 151–160, 2000     

Color-critical graphs have logarithmic circumference

A graph G is k-critical if every proper subgraph of G is (k−1)-colorable, but the graph G itself is not. We prove that every k-critical graph on n vertices has a cycle of length at least , improving a bound of Alon, Krivelevich and Seymour from 2000. Examples of Gallai from 1963 show that the bound cannot be improved to exceed . We thus settle the problem of bounding the minimal circumference of k-critical graphs, raised by Dirac in 1952 and Kelly and Kelly in 1954.     

关于偶正则图的R2边连通度的某些结果

Let G be a connected k(≥3)-regular graph with girth g. A set S of the edges in G is called an R2-edge-cut if G-S is disconnected and contains neither an isolated vertex nor a one-degree vertex. The R2-edge-connectivity of G, denoted by λ″(G), is the minimum cardinality over all R2-edge-cuts, which is an important measure for fault-tolerance of computer interconnection networks. In this paper, λ″(G)=g(2k-2) for any 2k-regular connected graph G(≠K5) that is either edge-transitive or vertex-transitive and g≥5 is given.     

Connectivity of Cartesian product graphs

Use v_i,κ_i,λ_i,δ_i to denote order, connectivity, edge-connectivity and minimum degree of a graph G_i for i=1,2, respectively. For the connectivity and the edge-connectivity of the Cartesian product graph, up to now, the best results are κ(G₁×G₂)?κ₁+κ₂ and λ(G₁×G₂)?λ₁+λ₂. This paper improves these results by proving that κ(G₁×G₂)?min{κ₁+δ₂,κ₂+δ₁} and λ(G₁×G₂)=min{δ₁+δ₂,λ₁v₂,λ₂v₁} if G₁ and G₂ are connected undirected graphs; κ(G₁×G₂)?min{κ₁+δ₂,κ₂+δ₁,2κ₁+κ₂,2κ₂+κ₁} if G₁ and G₂ are strongly connected digraphs. These results are also generalized to the Cartesian products of connected graphs and n strongly connected digraphs, respectively.     

Strong structural properties of unidirectional star graphs

Day and Tripathi [K. Day, A. Tripathi, Unidirectional star graphs, Inform. Process. Lett. 45 (1993) 123-129] proposed an assignment of directions on the star graphs and derived attractive properties for the resulting directed graphs: an important one is that they are strongly connected. In [E. Cheng, M.J. Lipman, On the Day-Tripathi orientation of the star graphs: Connectivity, Inform. Process. Lett. 73 (2000) 5-10] it is shown that the Day-Tripathi orientations are in fact maximally arc-connected when n is odd; when n is even, they can be augmented to maximally arc-connected digraphs by adding a minimum set of arcs. This gives strong evidence that the Day-Tripathi orientations are good orientations. In [E. Cheng, M.J. Lipman, Connectivity properties of unidirectional star graphs, Congr. Numer. 150 (2001) 33-42] it is shown that vertex-connectivity is maximal, and that if we delete as many vertices as the connectivity, we can create at most two strong connected components, at most one of which is not a singleton. In this paper we prove an asymptotically sharp upper bound for the number of vertices we can delete without creating two nonsingleton strong components, and we also give sharp upper bounds on the number of singletons that we might create.     

On the diameter of Kronecker graphs

We prove that a.a.s. as soon as a Kronecker graph becomes connected it has a finite diameter.     

On graph equivalences preserved under extensions

Let G be the set of finite graphs whose vertices belong to some fixed countable set, and let ≡ be an equivalence relation on G. By the strengthening of ≡ we mean an equivalence relation ≡_s such that G≡_sH, where G,H∈G, if for every F∈G, G∪F≡H∪F. The most important case that we study in this paper concerns equivalence relations defined by graph properties. We write G≡^ΦH, where Φ is a graph property and G,H∈G, if either both G and H have the property Φ, or both do not have it. We characterize the strengthening of the relations ≡^Φ for several graph properties Φ. For example, if Φ is the property of being a k-connected graph, we find a polynomially verifiable (for k fixed) condition that characterizes the pairs of graphs equivalent with respect to . We obtain similar results when Φ is the property of being k-colorable, edge 2-colorable, Hamiltonian, or planar, and when Φ is the property of containing a subgraph isomorphic to a fixed graph H. We also prove several general theorems that provide conditions for ≡_s to be of some specific form. For example, we find a necessary and sufficient condition for the relation ≡_s to be the identity. Finally, we make a few observations on the strengthening in a more general case when G is the set of finite subsets of some countable set.     

通过增强型萨格勒布指数对图排序

Recently, Furtula et al. proposed a valuable predictive index in the study of the heat of formation in octanes and heptanes, the augmented Zagreb index(AZI index) of a graph G, which is defined as AZI(G) =∑uv∈E(G)( d_u d_v/d_u + d_v-2)~3,where E(G) is the edge set of G, d u and d v are the degrees of the terminal vertices u and v of edge uv, respectively. In this paper, we obtain the first five largest(resp., the first two smallest) AZI indices of connected graphs with n vertices. Moreover, we determine the trees of order n with the first three smallest AZI indices, the unicyclic graphs of order n with the minimum, the second minimum AZI indices, and the bicyclic graphs of order n with the minimum AZI index, respectively.     

关于直径为4的整树

给出了直径为4的整树的一个实用的充要条件,讨论了几个基本的公开问题, 并给出了直径为4的整树的一些新类.     

直径为4的整树

本文给出了直径为4的整树的一个充分条件,并由此给出了直径为4的整树的许多新类．最后提出一些基本的公开问题．     

Harmonic trees

A graph G is defined to be harmonic if there is a constant λ (necessarily a natural number) such that, for every vertex v, the sum of the degrees of the neighbors of v equals λd_G (v) where d_G (v) is the degree of v. We show that there is exactly one finite harmonic tree for every λ ε , and we give a recursive construction for all infinite harmonic trees.