Infinite Eulerian paths are computable on graphs with vertices of infinite degree

A one-way infinite Hamiltonian path is constructed in an infinite 4-connected VAP-free maximal planar graph containing one or two vertices of infinite degree. Combining this result and that of R. HALIN who investigated the structure of such graphs, we conclude that such a path always exists in every infinite 4-connected maximal planar graph with exactly one end, which is an extension of H. WHITNEY'S theorem to infinite graphs.

Unavoidable induced subgraphs of infinite 2-connected graphs

For a graph A and a positive integer n, let nA denote the union of n disjoint copies of A; similarly, the union of ℵ0 disjoint copies of A is referred to as ℵ0A. It is shown that there exist (connected) graphs A and G such that nA is a minor of G for all nϵℕ, but ℵ0A is not a minor of G. This supplements previous examples showing that analogous statements are true if, instead of minors, isomorphic embeddings or topological minors are considered. The construction of A and G is based on the fact that there exist (infinite) graphs G1, G2,… such that Gi is not a minor of Gj for all i ≠ j. In contrast to previous examples concerning isomorphic embeddings and topological minors, the graphs A and G presented here are not locally finite. The following conjecture is suggested: for each locally finite connected graph A and each graph G, if nA is a minor of G for all n ϵ ℕ, then ℵ0A is a minor of G, too. If true, this would be a far‐reaching generalization of a classical result of R. Halin on families of disjoint one‐way infinite paths in graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 222–229, 2002; DOI 10.1002/jgt.10016

This paper considers the feasibility of finite and infinite paths in programs in two simple programming languages. The language LBASE allows to express the dependencies of real time systems on integer data, the language LTIM can model quantitative timing constraints in r.t.s. specifications. It is proven that the problem of whether a given LBASE or LTIM program has an infinite feasible path (i.e. whether it can exhibit an infinite behaviour) is decidable. The possibilities to characterise the sets of all feasible finite and infinite paths in LBASE and LTIM programs are also discussed. The infinite feasible path existence problem is proven decidable also for the language LTIBA which has both the LBASE and LTIM modelling capabilities.

A class of [formula omitted]-perfect graphs

In this chapter we discuss countably infinite connected simple graphs that are locally finite , that is, the vertex degrees are finite. In a similar fashion to the previous chapter, an infinite planar graph is a connected infinite graph such that there exists a drawing of it in the plane. We recall that a drawing is a correspondence sending vertices to points of \(\mathbb {R}^2\) and edges to continuous curves between the corresponding vertices such that no two edges cross. An infinite planar map is an infinite planar graph equipped with a set of cyclic permutations {σv : v ∈ V } of the neighbors of each vertex v, such that there exists a drawing of the graph which respects these permutations, that is, the clockwise order of edges emanating from a vertex v coincides with σv.

Optimal identifying codes of two families of Cayley graphs

Decomposition of infinite eulerian graphs with a small number of vertices of infinite degree

It is a well established fact, that—in the case of classical random graphs like (variants of) G n,p or random regular graphs-spectral methods yield efficient algorithms for clustering (e. g. coiouring or bisection) problems. The theory of large networks emerging recently provides convincing evidence that such networks, albeit looking random in some sense, cannot sensibly be described by classical random graphs. A variety of new types of random graphs have been introduced. One of these types is characterized by the fact that we have a fixed expected degree sequence, that is for each vertex its expected degree is given. Recent theoretical work confirms that spectral methods can be successfully applied to clustering problems for such random graphs, too—provided that the expected degrees are not too small, in fact≥log6 n. In this case however the degree of each vertex is concentrated about its expectation. We show how to remove this restriction and apply spectral methods when the expected degrees are bounded below just by a suitable constant. Our results rely on the observation that techniques developed for the classical sparse G n,p random graph (that is p=c/n) can be transferred to the present situation, when we consider a suitably normalized adjacency matrix: We divide each entry of the adjacency matrix by the product of the expected degrees of the incident vertices. Given the host of spectral techniques developed for G n,p this observation should be of independent interest.

It is shown that the discharging method can be successfully applied on infinite planar graphs of subexponential growth and even on those graphs that do not satisfy the strong edge isoperimetric inequality. The general outline of the method is presented and the following applications are given: Planar graphs with only finitely many vertices of degree ≤ 5 and with subexponential growth contain arbitrarily large finite submaps of the tessellation of the plane or of some tessellation of the cylinder by equilateral triangles. Every planar graph with isoperimetric number zero and with essential minimum degree > 3 has infinitely many edges whose degree sum is at most 15. In particular, this holds for all graphs with minimum degree > 3 and with subexponential growth. The cases without infinitely many edges whose degree sum is < 14 (or, similarly, < 13 or < 12) are also considered. Several further results are obtained.

A theorem of Mader states that highly connected subgraphs can be forced in finite graphs by assuming a high minimum degree. We extend this result to infinite graphs. Here, it is necessary to require not only high degree for the vertices but also high vertex‐degree (or multiplicity) for the ends of the graph, that is, a large number of disjoint rays in each end. We give a lower bound on the degree of vertices and the vertex‐degree of the ends which is quadratic in k, the connectedness of the desired subgraph. In fact, this is not far from best possible: we exhibit a family of graphs with a degree of order 2k at the vertices and a vertex‐degree of order k log k at the ends which have no k‐connected subgraphs.Furthermore, if in addition to the high degrees at the vertices, we only require high edge‐degree for the ends (which is defined as the maximum number of edge‐disjoint rays in an end), Mader's theorem does not extend to infinite graphs, not even to locally finite ones. We give a counterexample in this respect. But, assuming a lower bound of at least 2k for the edge‐degree at the ends and the degree at the vertices does suffice to ensure the existence (k + 1)‐edge‐connected subgraphs in arbitrary graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 331–349, 2007

The property of spatial mixing and strong spatial mixing in spin systems has been of interest because of its implications on uniqueness of Gibbs measures on infinite graphs and efficient approximation of counting problems that are otherwise known to be #P hard. In the context of coloring, strong spatial mixing has been established for Kelly trees in (Ge and Stefankovic, arXiv:1102.2886v3 (2011)) when where q the number of colors, Δ is the degree and .. is the unique solution to . It has also been established in (Goldberg et al., SICOMP 35 (2005) 486–517) for bounded degree lattice graphs whenever for some constant β, where Δ is the maximum vertex degree of the graph. We establish strong spatial mixing for a more general problem, namely list coloring, for arbitrary bounded degree triangle‐free graphs. Our results hold for any whenever the size of the list of each vertex v is at least where is the degree of vertex v and β is a constant that only depends on α. The result is obtained by proving the decay of correlations of marginal probabilities associated with graph nodes measured using a suitably chosen error function. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46,599–613, 2015

We study determinacy, definability and complexity issues of path games on finite and infinite graphs. Compared to the usual format of infinite games on graphs (such as Gale-Stewart games) we consider here a different variant where the players select in each move a path of arbitrary finite length, rather than just an edge. The outcome of a play is an infinite path, the winning condition hence is a set of infinite paths, possibly given by a formula from S1S, LTL, or first-order logic. Such games have a long tradition in descriptive set theory (in the form of Banach-Mazur games) and have recently been shown to have interesting application for planning in nondeterministic domains. It turns out that path games behave quite differently than classical graph games. For instance, path games with Muller conditions always admit positional winning strategies which are computable in polynomial time. With any logic on infinite paths (defining a winning condition) we can associate a logic on graphs, defining the winning regions of the associated path games. We explore the relationships between these logics. For instance, the winning regions of path games with an S1S-winning condition are definable in the modal mu-calculus. Further, if the winning condition is first-order (on paths), then the winning regions are definable in monadic path logic, or, for a large class of games, even in first-order logic. As a consequence, winning regions of LTL path games are definable in CTL.

A class C of graphs is said to be dually compact closed if, for every infinite G ∈ C, each finite subgraph of G is contained in a finite induced subgraph of G which belongs to C. The class of trees and more generally the one of chordal graphs are dually compact closed. One of the main part of this paper is to settle a question of Hahn, Sands, Sauer and Woodrow by showing that the class of bridged graphs is dually compact closed. To prove this result we use the concept of constructible graph. A (finite or infinite) graph G is constructible if there exists a wellordering ≤ (called constructing ordering) of its vertices such that, for every vertex x which is not the smallest element, there is a vertex y < x which is adjacent to x and to every neighbor z of x with z < x. Finite graphs are constructible if and only if they are dismantlable. The case is different, however, with infinite graphs. A graph G for which every breadth-first search of G produces a particular constructing ordering of its vertices is called a BFS-constructible graph. We show that the class of BFS-constructible graphs is a variety (i.e., it is closed under weak retracts and strong products), that it is a subclass of the class of weakly modular graphs, and that it contains the class of bridged graphs and that of Helly graphs (bridged graphs being very special instances of BFS-constructible graphs). Finally we show that the class of intervalfinite pseudo-median graphs (and thus the one of median graphs) and the class of Helly graphs are dually compact closed, and that moreover every finite subgraph of an interval-finite pseudo-median graph (resp. a Helly graph) G is contained in a finite isometric pseudo-median

A graph is k‐indivisible, where k is a positive integer, if the deletion of any finite set of vertices results in at most k – 1 infinite components. In 1971, Nash‐Williams conjectured that a 4‐connected infinite planar graph contains a spanning 2‐way infinite path if and only if it is 3‐indivisible. In this paper, we prove a structural result for 2‐indivisible infinite planar graphs. This structural result is then used to prove Nash‐Williams conjecture for all 4‐connected 2‐indivisible infinite planar graphs. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 247–266, 2005