Finite Paths are Universal

We prove that any countable (finite or infinite) partially ordered set may be represented by finite oriented paths ordered by the existence of homomorphism between them. This (what we believe a surprising result) solves several open problems. Such path-representations were previously known only for finite and infinite partial orders of dimension 2. Path-representation implies the universality of other classes of graphs (such as connected cubic planar graphs). It also implies that finite partially ordered sets are on-line representable by paths and their homomorphisms. This leads to new on-line dimensions. Mathematics Subject Classifications (2000) 06A06, 06A07, 05E99, 05C99.J. Nešetřil: Supported by a Grant LN00A56 of the Czech Ministry of Education. The first author was partially supported by EU network COMBSTRU at UPC Barcelona.     

A census of semisymmetric cubic graphs on up to 768 vertices

A list is given of all semisymmetric (edge- but not vertex-transitive) connected finite cubic graphs of order up to 768. This list was determined by the authors using Goldschmidt's classification of finite primitive amalgams of index (3,3), and a computer algorithm for finding all normal subgroups of up to a given index in a finitely-presented group. The list includes several previously undiscovered graphs. For each graph in the list, a significant amount of information is provided, including its girth and diameter, the order of its automorphism group, the order and structure of a minimal edge-transitive group of automorphisms, its Goldschmidt type, stabiliser partitions, and other details about its quotients and covers. A summary of all known infinite families of semisymmetric cubic graphs is also given, together with explicit rules for their construction, and members of the list are identified with these. The special case of those graphs having K_1,3 as a normal quotient is investigated in detail. Supported in part by N.Z. Marsden Fund (grant no. UOA 124) and N.Z. Centres of Research Excellence Fund (grant no. UOA 201) Supported in part by “Ministrstvo za šolstvo, znanost in šport Slovenije”, research program no. 101-506. Supported in part by research projects no. Z1-4186-0101 and no. Z1-3124-0101. The fourth author would like to thank the University of Auckland for hospitality during his visit there in 2003.     

Two-ended regular median graphs

We show that regular median graphs of linear growth are the Cartesian product of finite hypercubes with the two-way infinite path. Such graphs are Cayley graphs and have only two ends.For cubic median graphs G the condition of linear growth can be weakened to the condition that G has two ends. For higher degree the relaxation to two-ended graphs is not possible, which we demonstrate by an example of a median graph of degree four that has two ends, but nonlinear growth.     

Self-Complementary Vertex-Transitive Graphs Need Not be Cayley Graphs

A construction is given of an infinite family of finite self-complementary,vertex-transitive graphs which are not Cayley graphs. To theauthors' knowledge, these are the first known examples of suchgraphs. The nature of the construction was suggested by a generalstudy of the structure of self-complementary, vertex-transitivegraphs. It involves the product action of a wreath product ofpermutation groups. 2000 Mathematics Subject Classification05C25.     

Uncountable families of vertex-transitive graphs of finite degree

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

Vertex-transitive graphs and vertex-transitive maps

We consider vertex-transitive graphs embeddable on a fixed surface. We prove that all but a finite number of them admit embeddings as vertex-transitive maps on surfaces of nonnegative Euler characteristic (sphere, projective plane, torus, or Klein bottle). It follows that with the exception of the cycles and a finite number of additional graphs, they are factor graphs of semiregular plane tilings. The results generalize previous work on the genus of minimal Cayley graphs by V. Proulx and T. W. Tucker and were obtained independently by C. Thomassen, with significant differences in the methods used. Our method is based on an excursion into the infinite. The local structure of our finite graphs is studied via a pointwise limit construction, and the infinite vertex-transitive graphs obtained as such limits are classified by their connectivity and the number of ends. In two appendices, we derive a combinatorial version of Hurwitz's Theorem, and classify the vertex-transitive maps on the Klein bottle.     

A Note on Adjacency-Transitivity of a Graph and Its Complement

We give a necessary and sufficient condition for a graphical regular representation to be adjacency-transitive, and provide an infinite family of finite simple undirected vertex-transitive graphs Γ, such that neither Γ nor Γ ^c is adjacency-transitive. Revised: March 24, 1998     

Some Elementary Ramanujan Graphs

We give elementary constructions of two infinite families of Ramanujan graphs of unbounded degree. The first uses the geometry of buildings over finite fields, and the second uses triangulations of modular curves.Mathematics Subject Classiffications (2000). Primary: 05C25; secondary: 05C50, 51E24     

A note on fragments of infinite graphs

Results involving automorphisms and fragments of infinite graphs are proved. In particular for a given fragmentC and a vertex-transitive subgroupG of the automorphism group of a connected graph there exists σ≠G such that σ[C] ⊂C. This proves the countable case of a conjecture of L. Babai and M. E. Watkins concerning graphs allowing a vertex-transitive torsion group action. Dedicated to Prof. K. Wagner on his 70^th birthday     

Edge-transitive products

This paper concerns finite, edge-transitive direct and strong products, as well as infinite weak Cartesian products. We prove that the direct product of two connected, non-bipartite graphs is edge-transitive if and only if both factors are edge-transitive and at least one is arc-transitive, or one factor is edge-transitive and the other is a complete graph with loops at each vertex. Also, a strong product is edge-transitive if and only if all factors are complete graphs. In addition, a connected, infinite non-trivial Cartesian product graph G is edge-transitive if and only if it is vertex-transitive and if G is a finite weak Cartesian power of a connected, edge- and vertex-transitive graph H, or if G is the weak Cartesian power of a connected, bipartite, edge-transitive graph H that is not vertex-transitive.     

A note on vertex-transitive non-Cayley graphs from Cayley graphs generated by involutions

We show that the result of Watkins (1990) [19] on constructing vertex-transitive non-Cayley graphs from line graphs yields a simple method that produces infinite families of vertex-transitive non-Cayley graphs from Cayley graphs generated by involutions. We also prove that the graphs arising this way are hamiltonian provided that their valency is at least six.     

On the edge-connectivity of graphs with two orbits of the same size

It is well known that the edge-connectivity of a simple, connected, vertex-transitive graph attains its regular degree. It is then natural to consider the relationship between the graph’s edge-connectivity and the number of orbits of its automorphism group. In this paper, we discuss the edge connectedness of graphs with two orbits of the same size, and characterize when these double-orbit graphs are maximally edge connected and super-edge-connected. We also obtain a sufficient condition for some double-orbit graphs to be λ^′-optimal. Furthermore, by applying our results we obtain some results on vertex/edge-transitive bipartite graphs, mixed Cayley graphs and half vertex-transitive graphs.     

The vertex-transitive TLF-planar graphs

We consider the class of the topologically locally finite (in short TLF) planar vertex-transitive graphs. We characterize these graphs by finite combinatorial objects called labeling schemes. As a result, we are able to enumerate and describe all TLF-planar vertex-transitive graphs of given degree, as well as most of their transitive groups of automorphisms. In addition,we are able to decide whether a given TLF-planar transitive graph is Cayley or not. This class contains all the one-ended planar Cayley graphs and the normal transitive tilings of the plane.     

Super cyclically edge-connected vertex-transitive graphs of girth at least 5

A cyclic edge-cut of a graph G is an edge set, the removal of which separates two cycles. If G has a cyclic edge-cut, then it is called cyclically separable. We call a cyclically separable graph super cyclically edge-connected, in short, super-λc, if the removal of any minimum cyclic edge-cut results in a component which is a shortest cycle. In [Zhang, Z., Wang, B.: Super cyclically edge-connected transitive graphs. J. Combin. Optim., 22, 549-562 (2011)], it is proved that a connected vertex-transitive graph is super-λc if G has minimum degree at least 4 and girth at least 6, and the authors also presented a class of nonsuper-λc graphs which have degree 4 and girth 5. In this paper, a characterization of k (k≥4)-regular vertex-transitive nonsuper-λc graphs of girth 5 is given. Using this, we classify all k (k≥4)-regular nonsuper-λc Cayley graphs of girth 5, and construct the first infinite family of nonsuper-λc vertex-transitive non-Cayley graphs.     

Some Recent Progress and Applications in Graph Minor Theory

In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the ``rough' structure of graphs excluding a fixed minor. This result was used to prove Wagner's Conjecture that finite graphs are well-quasi-ordered under the graph minor relation. Recently, a number of beautiful results that use this structural result have appeared. Some of these along with some other recent advances on graph minors are surveyed. Research partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, Grant number 16740044, by Sumitomo Foundation, by C & C Foundation and by Inoue Research Award for Young Scientists Supported in part by the Research Grant P1–0297 and by the CRC program On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia     

Homogeneous factorisations of Johnson graphs

For a graph Γ, subgroups , and an edge partition of Γ, the pair is a (G, M)-homogeneous factorisation if M is vertex-transitive on Γ and fixes setwise each part of , while G permutes the parts of transitively. A classification is given of all homogeneous factorisations of finite Johnson graphs. There are three infinite families and nine sporadic examples. This paper forms part of an ARC Discovery grant of the last two authors. The second author holds an Australian Research Council Australian Research Fellowship.     

Hamiltonian paths in infinite graphs

A tight connection is exhibited between infinite paths in recursive trees and Hamiltonian paths in recursive graphs. A corollary is that determining Hamiltonicity in recursive graphs is highly undecidable, viz, Σ ₁ ¹ -complete. This is shown to hold even for highly recursive graphs with degree bounded by 3. Hamiltonicity is thus an example of an interesting graph problem that is outside the arithmetic hierarchy in the infinite case. Parts of this research were carried out during a visit to IBM T.J. Watson Research Center, Hawthorne, NY, in the Summer of 1990. The author holds the William Sussman Professorial Chair in Mathematics.     

Enumerating Planar Locally Finite Cayley Graphs

We characterize the set of planar locally finite Cayley graphs, and give a finite representation of these graphs by a special kind of finite state automata called labeling schemes. As a result, we are able to enumerate and describe all planar locally finite Cayley graphs of a given degree. This analysis allows us to solve the problem of decision of the locally finite planarity for a word-problem-decidable presentation.Mathematics Subject Classiffications (2000). 20F05, 20F10, 20F65, 05C25     

Rectangular and visibility representations of infinite planar graphs

We provide a new method for extending results on finite planar graphs to the infinite case. Thus a result of Ungar on finite graphs has the following extension: Every infinite, planar, cubic, cyclically 4‐edge‐connected graph has a representation in the plane such that every edge is a horizontal or vertical straight line segment, and such that no two edges cross. A result of Tamassia and Tollis extends as follows: Every countably infinite planar graph is a subgraph of a visibility graph. Furthermore, every locally finite, 2‐connected, planar graph is a visibility graph. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 257–265, 2006