Characterizations of bipartite and Eulerian partial duals of ribbon graphs

Huggett and Moffatt characterized all bipartite partial duals of a plane graph in terms of all-crossing directions of its medial graph. Then Metsidik and Jin characterized all Eulerian partial duals of a plane graph in terms of semi-crossing directions of its medial graph. Plane graphs are ribbon graphs with genus 0. In this paper, by introducing the notion of modified medial graphs and using their all-crossing directions, we first extend Huggett and Moffatt’s result from plane graphs to ribbon graphs. Then we characterize all Eulerian partial duals of any ribbon graph in terms of crossing-total directions of its medial graph, which are simpler than semi-crossing directions.     

Locally finite graphs with ends: A topological approach, I. Basic theory

This paper is the first of three parts of a comprehensive survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. The first two parts of the survey together provide a suitable entry point to this field for new readers; they are available in combined form from the ArXiv [18]. They are complemented by a third part [28], which looks at the theory from an algebraic-topological point of view.The topological approach indicated above has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. While the second part of this survey [19] will concentrate on those applications, this first part explores the new theory as such: it introduces the basic concepts and facts, describes some of the proof techniques that have emerged over the past 10 years (as well as some of the pitfalls these proofs have in stall for the naive explorer), and establishes connections to neighbouring fields such as algebraic topology and infinite matroids. Numerous open problems are suggested.     

Locally finite graphs with ends: A topological approach, II. Applications

This paper is the second of three parts of a comprehensive survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. The first two parts of the survey together provide a suitable entry point to this field for new readers; they are available in combined form from the ArXiv [20]. They are complemented by a third part [31], which looks at the theory from an algebraic-topological point of view.The topological approach indicated above has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. This second part of the survey concentrates on these applications, many of which solve problems or extend earlier work of Thomassen on infinite graphs. Numerous new problems are suggested.     

Eulerian and even-face ribbon graph minors

In this paper, we introduce Eulerian and even-face ribbon graph minors. These minors preserve Eulerian and even-face properties of ribbon graphs, respectively. We then characterize Eulerian, even-face, plane Eulerian and plane even-face ribbon graphs using these minors.     

欧拉图与矩阵环的多项式恒等式

本文运用Swan证明Amitsur-levitzki定理所用有向路图论方法，获得了交换环上矩阵环所满足的一类新型多项式恒等式，标准多项式恒等式和Chang-Giambruno-Sehgal多项式恒等式是我们所得恒等式的特例。     

Asymptotic enumeration theorems for the numbers of spanning trees and Eulerian trails in circulant digraphs and graphs

The asymptotic properties of the numbers of spanning trees and Eulerian trails in circulant digraphs and graphs are studied. Let be a directed circulant graph. Let and be the numbers of spanning trees and of Eulerian trails, respectively. Then

Eulerian subgraphs in 3‐edge‐connected graphs and Hamiltonian line graphs

In this paper, we show that if G is a 3‐edge‐connected graph with and , then either G has an Eulerian subgraph H such that , or G can be contracted to the Petersen graph in such a way that the preimage of each vertex of the Petersen graph contains at least one vertex in S. If G is a 3‐edge‐connected planar graph, then for any , G has an Eulerian subgraph H such that . As an application, we obtain a new result on Hamiltonian line graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 308–319, 2003     

Ends,tangles and critical vertex sets

We show that an arbitrary infinite graph G can be compactified by its ends plus its critical vertex sets, where a finite set X of vertices of an infinite graph is critical if its deletion leaves some infinitely many components each with neighbourhood precisely equal to X. We further provide a concrete separation system whose ?₀‐tangles are precisely the ends plus critical vertex sets. Our tangle compactification is a quotient of Diestel's (denoted by ), and both use tangles to compactify a graph in much the same way as the ends of a locally finite and connected graph compactify it in its Freudenthal compactification. Finally, generalising both Diestel's construction of and our construction of , we show that G can be compactified by every inverse limit of compactifications of the sets of components obtained by deleting a finite set of vertices. Diestel's is the finest such compactification, and our is the coarsest one. Both coincide if and only if all tangles are ends. This answers two questions of Diestel.     

Circumference of 3-connected claw-free graphs and large Eulerian subgraphs of 3-edge-connected graphs

The circumference of a graph is the length of its longest cycles. Results of Jackson, and Jackson and Wormald, imply that the circumference of a 3-connected cubic n-vertex graph is Ω(n^0.694), and the circumference of a 3-connected claw-free graph is Ω(n^0.121). We generalize and improve the first result by showing that every 3-edge-connected graph with m edges has an Eulerian subgraph with Ω(m^0.753) edges. We use this result together with the Ryjá?ek closure operation to improve the lower bound on the circumference of a 3-connected claw-free graph to Ω(n^0.753). Our proofs imply polynomial time algorithms for finding large Eulerian subgraphs of 3-edge-connected graphs and long cycles in 3-connected claw-free graphs.     

Analysis of the physical Laplacian and the heat flow on a locally finite graph

We study the physical Laplacian and the corresponding heat flow on an infinite, locally finite graph with possibly unbounded valence.     

Graph-like spaces: An introduction

Thomassen and Vella (Graph-like continua, augmenting arcs, and Menger’s Theorem, Combinatorica, doi:10.1007/s00493-008-2342-9) have recently introduced the notion of a graph-like space, simultaneously generalizing infinite graphs and many of the compact spaces recently used by Diestel or Richter (and their coauthors) to study cycle spaces and related problems in infinite graphs. This work is a survey to introduce graph-like spaces and shows how many of these works on compact spaces can be generalized to compact graph-like spaces.     

Normal Vietoris Implies Compactness: A Short Proof

One of the most celebrated results in the theory of hyperspaces says that if the Vietoris topology on the family of all nonempty closed subsets of a given space is normal, then the space is compact (Ivanova-Keesling-Velichko). The known proofs use cardinality arguments and are long. In this paper we present a short proof using known results concerning Hausdorff uniformities.     

Characterizations of universal finite representability and b-convexity of Banach spaces via ball coverings

By a ball-covering B of a Banach space X, wemean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere of X; and X is said to have the ball-covering property provided it admits a ball-covering of countably many balls. This paper shows that universal finite representability and B-convexity of X can be characterized by properties of ball-coverings of its finite dimensional subspaces.     

Notions of Local Compactness and Smallest Compactifications of Biframes

The category of coherent biframes is shown to be equivalent to that of the coupled lattices, and dually equivalent to the spectral bispaces. Stably continuous biframes arise as the retracts of the coherent biframes. The coherent and the stably continuous biframes are coreflective in all biframes. Weak local compactness is introduced, and in conjunction with regularity, is shown to be sufficient for the construction of smallest compactifications.     

Between ends and fibers

Let Γ be an infinite, locally finite, connected graph with distance function δ. Given a ray P in Γ and a constant C ≥ 1, a vertex‐sequence is said to be regulated by C if, for all n??, never precedes x_n on P, each vertex of P appears at most C times in the sequence, and . R. Halin (Math. Ann., 157, 3 , 125–137) defined two rays to be end‐equivalent if they are joined by infinitely many pairwise‐disjoint paths; the resulting equivalence classes are called ends. More recently H. A. Jung (Graph Structure Theory, Contemporary Mathematics, 147, 6 , 477–484) defined rays P and Q to be b‐equivalent if there exist sequences and VQ regulated by some constant C ≥ 1 such that for all n??; he named the resulting equivalence classes b‐fibers. Let denote the set of nondecreasing functions from into the set of positive real numbers. The relation (called f‐equivalence) generalizes Jung's condition to . As f runs through , uncountably many equivalence relations are produced on the set of rays that are no finer than b‐equivalence while, under specified conditions, are no coarser than end‐equivalence. Indeed, for every Γ there exists an “end‐defining function” that is unbounded and sublinear and such that implies that P and Q are end‐equivalent. Say if there exists a sublinear function such that . The equivalence classes with respect to are called bundles. We pursue the notion of “initially metric” rays in relation to bundles, and show that in any bundle either all or none of its rays are initially metric. Furthermore, initially metric rays in the same bundle are end‐equivalent. In the case that Γ contains translatable rays we give some sufficient conditions for every f‐equivalence class to contain uncountably many g‐equivalence classes (where ). We conclude with a variety of applications to infinite planar graphs. Among these, it is shown that two rays whose union is the boundary of an infinite face of an almost‐transitive planar map are never bundle‐ equivalent. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 125–153, 2007     

Geodetic rays and fibers in periodic graphs

Using the notion of fibers, where two rays belong to the same fiber if and only if they lie within bounded Hausdorff‐distance of one another, we study how many fibers of a graph contain a geodetic ray and how many essentially distinct geodetic rays such “geodetic fibers” must contain. A complete answer is provided in the case of locally finite graphs that admit an almost transitive action by some infinite finitely generated, abelian group. Such graphs turn out to have either finitely many or uncountably many geodetic fibers. Furthermore, with finitely many possible exceptions, each of these fibers contains uncountably many geodetic rays. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 67–88, 2000     

Global gradient estimate on graph and its applications

Continuing our previous work(ar Xiv:1509.07981v1),we derive another global gradient estimate for positive functions,particularly for positive solutions to the heat equation on finite or locally finite graphs.In general,the gradient estimate in the present paper is independent of our previous one.As applications,it can be used to get an upper bound and a lower bound of the heat kernel on locally finite graphs.These global gradient estimates can be compared with the Li–Yau inequality on graphs contributed by Bauer et al.[J.Differential Geom.,99,359–409(2015)].In many topics,such as eigenvalue estimate and heat kernel estimate(not including the Liouville type theorems),replacing the Li–Yau inequality by the global gradient estimate,we can get similar results.