Reconstruction of infinite locally finite connected graphs

Let G be an infinite locally finite connected graph. We study the reconstructibility of G in relation to the structure of its end set . We prove that an infinite locally finite connected graph G is reconstructible if there exists a finite family (Ω_i)_0i (n2) of pairwise finitely separable subsets of such that, for all x,y,x′,y′V(G) and every isomorphism f of G−{x,y} onto G−{x′,y′} there is a permutation π of {0,…,n−1} such that for 0i<n. From this theorem we deduce, as particular consequences, that G is reconstructible if it satisfies one of the following properties: (i) G contains no end-respecting subdivision of the dyadic tree and has at least two ends of maximal order; (ii) the set of thick ends or the one of thin ends of G is finite and of cardinality greater than one. We also prove that if almost all vertices of G are cutvertices, then G is reconstructible if it contains a free end or if it has at least a vertex which is not a cutvertex.     

Uncountable families of vertex-transitive graphs of finite degree

We prove that the set of vertex-transitive graphs of finite degree is uncountably large.     

Value-true walks in finite,connected, valuated graphs

Starting from Problem 4 on p. 36 ofK. Wagner's book “Graphentheorie” the following question is investigated: Suppose an (undirected) graphG having its edges valuated with nonnegative integers; does there exist a closed walk inG such that each 0-valuated edge is passed the same number of times in both directions, while ak-valuated edge fork>0, is passed exactlyk times in one direction and not at all in the other direction (without having the direction prescribed)? A general criterion and a sufficient condition for the existence of such “value-true walks” are proved. For the analogous question for directed graphs an algorithm for the construction of such walks is established.     

Reconstruction of locally finite connected graphs with at least three infinite wings

Let G be a locally finite connected graph that can be expressed as the union of a finite subgraph and p disjoint infinite subgraphs, where 3 ≦ p < ∞, but cannot be expressed as the union of a finite subgraph and p + 1 disjoint infinite subgraphs. Then G is reconstructible.     

Shift graphs on precompact families of finite sets of natural numbers

We study graphs defined on families of finite sets of natural numbers and their chromatic properties. Of particular interest are graphs for which the edge relation is given by the shift. We show that when considering shift graphs with infinite chromatic number, one can center attention on graphs defined on precompact thin families. We define a quasi-order relation on the collection of uniform families defined in terms of homomorphisms between their corresponding shift graphs, and show that there are descending ${\omega }_1$-sequences. Specker graphs are also considered and their relation with shift graphs is established. We characterize the family of Specker graphs which contain a homomorphic image of a shift graph.     

Completely connected clustered graphs

Planar drawings of clustered graphs are considered. We introduce the notion of completely connected clustered graphs, i.e., hierarchically clustered graphs that have the property that not only every cluster but also each complement of a cluster induces a connected subgraph. As a main result, we prove that a completely connected clustered graph is c-planar if and only if the underlying graph is planar. Further, we investigate the influence of the root of the inclusion tree to the choice of the outer face of the underlying graph and vice versa.     

On spanning connected graphs

A k-containerC(u,v) of G between u and v is a set of k internally disjoint paths between u and v. A k-container C(u,v) of G is a k^*-container if the set of the vertices of all the paths in C(u,v) contains all the vertices of G. A graph G is k^*-connected if there exists a k^*-container between any two distinct vertices. Therefore, a graph is 1^*-connected (respectively, 2^*-connected) if and only if it is hamiltonian connected (respectively, hamiltonian). In this paper, a classical theorem of Ore, providing sufficient conditional for a graph to be hamiltonian (respectively, hamiltonian connected), is generalized to k^*-connected graphs.     

On path connected graphs

Given n and i, n > 2, 2 ≤ i ≤ n ? 1, the smallest size of an n-graph without endvertices is obtained, which ensures a path of length i between any two vertices of the graph.     

Counting connected graphs asymptotically

We find the asymptotic number of connected graphs with k vertices and k−1+l edges when k,l approach infinity, re-proving a result of Bender, Canfield and McKay. We use the probabilistic method, analyzing breadth-first search on the random graph G(k,p) for an appropriate edge probability p. Central is the analysis of a random walk with fixed beginning and end which is tilted to the left.     

Pancyclicity of connected circulant graphs

The circulant G_n(a₁, ⋖, a_k), where 0 < a₁ < ··· < a_k < (n + 1)/2, is defined as the vertex-transitive graph that has vertices i ± a₁, ···, i ± a_k(mod n) adjacent to each vertex i. In this work we show that the connected circulants of degree at least three contain all even cycles. In addition, we prove that the connected circulants of girth three contain cycles of all lengths. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 17–25, 1997     

Enumeration of connected invariant graphs

Let h be a finite group acting on unlabeled graphs which does not change connectivity. Examples include edge reversal in directed graphs and permutations of colors in edge and/or vertex colored graphs. The generating functions of h-invariant (directed) graphs and h-invariant (weakly) connected (directed) graphs are discussed. This leads to a recursive formula for calculating the number of connected graphs when the total number of graphs is known. This is then applied to self-dual signed graphs, self-converse digraphs, and color cyclic graphs. Asymptotic expansions are also obtained. As expected, almost all of the above h-invariant graphs are connected and the asymptotic number of disconnected graphs has a simple interpretation.     

Quasisimple non-simple finite PC-groups exist

It is shown that quasisimple non-simple finite PC-groups exist.     

Unavoidable doubly connected large graphs

A connected graph is doubly connected if its complement is also connected. The following Ramsey-type theorem is proved in this paper. There exists a function h(n), defined on the set of integers exceeding three, such that every doubly connected graph on at least h(n) vertices must contain, as an induced subgraph, a doubly connected graph, which is either one of the following graphs or the complement of one of the following graphs:

(1) P_n, a path on n vertices;

(2) K_1,n^s, the graph obtained from K_1,n by subdividing an edge once;

(3) K_2,ne, the graph obtained from K_2,n by deleting an edge;

(4) K_2,n⁺, the graph obtained from K_2,n by adding an edge between the two degree-n vertices x₁ and x₂, and a pendent edge at each x_i.

Two applications of this result are also discussed in the paper.     

When are random graphs connected

Several results concerning the connectivity of infinite random graphs are considered. A necessary sufficient condition for a zero–one law to hold is given when the edges are chosen independently. Some specific examples are treated including one where the vertex set isN and the probability that an edge joiningi toj is present depends only on |i−j|.     

On perfectly two-edge connected graphs

This paper studies the graphs for which the 2-edge connected spanning subgraph polytope is completely described by the trivial inequalities and the so-called cut inequalities. These graphs are called perfectly 2-edge connected. The class of perfectly 2-edge connected graphs contains for instance the class of series-parallel graphs. We introduce a new class of perfectly 2-edge connected graphs. We discuss some structural properties of graphs which are (minimally with respect to some reduction operations) nonperfectly 2-edge connected. Using this we give sufficient conditions for a graph to be perfectly 2-edge connected.