A construction of an infinite family of 2-arc transitive polygonal graphs of arbitrary even girth

A near-polygonal graph is a graph Γ which has a set C\mathcal{C} of m-cycles for some positive integer m such that each 2-path of Γ is contained in exactly one cycle in C\mathcal{C}. If m is the girth of Γ, then the graph is called polygonal. We provide a construction of an infinite family of polygonal graphs of arbitrary even girth with 2-arc transitive automorphism groups, showing that there are infinitely many 2-arc transitive polygonal graphs of every girth.     

A construction of an infinite family of 2-arc transitive polygonal graphs of arbitrary odd girth

A near-polygonal graph is a graph Γ which has a set C of m-cycles for some positive integer m such that each 2-path of Γ is contained in exactly one cycle in C. If m is the girth of Γ then the graph is called polygonal. We provide a construction of an infinite family of polygonal graphs of arbitrary odd girth with 2-arc transitive automorphism groups.     

Finite 2‐Distance Transitive Graphs

A noncomplete graph Γ is said to be (G, 2)‐distance transitive if G is a subgroup of the automorphism group of Γ that is transitive on the vertex set of Γ, and for any vertex u of Γ, the stabilizer is transitive on the sets of vertices at distances 1 and 2 from u. This article investigates the family of (G, 2)‐distance transitive graphs that are not (G, 2)‐arc transitive. Our main result is the classification of such graphs of valency not greater than 5. We also prove several results about (G, 2)‐distance transitive, but not (G, 2)‐arc transitive graphs of girth 4.     

Symmetrical path-cycle covers of a graph and polygonal graphs

A near-polygonal graph is a graph Γ which has a set C of m-cycles for some positive integer m such that each 2-path of Γ is contained in exactly one cycle in C. If m is the girth of Γ then the graph is called polygonal. We introduce a method for constructing near-polygonal graphs with 2-arc transitive automorphism groups. As special cases, we obtain several new infinite families of polygonal graphs.     

A Family of Near-Polygonal Graphs of Valency 10

A near-polygonal graph is a graph ?? which has a set ${\mathcal{C}}$ of m-cycles for some positive integer m such that each 2-path of ?? is contained in exactly one cycle in ${\mathcal{C}}$ . If m is the girth of ?? then the graph is called polygonal. We provide a construction for an infinite family of 2-arc transitive near-polygonal graphs of valency 10 and provide a method for determining the length m of special cycles in graphs in this family. This provides us with some new instances of polygonal graphs as well.     

A note on the girth‐doubling construction for polygonal graphs

A near‐polygonal graph is a graph Γ which has a set ?? of m‐cycles for some positive integer m such that each 2‐path of Γ is contained in exactly one cycle in ??. If m is the girth of Γ then the graph is called polygonal. Given a polygonal graph Γ of valency r and girth m, Archdeacon and Perkel proved the existence of a polygonal graph Γ₂ of valency r and girth 2m. We will show that this construction can be extended to one that yields a polygonal graph Γ₃ of valency r and girth 3m, but that making the cycles any longer with this construction does not yield a polygonal graph. We also show that if Aut(Γ) is 2‐arc transitive, so is Aut(Γ_k) for k = 2, 3. © 2010 Wiley Periodicals, Inc. J Graph Theory 68: 246‐254, 2011     

Tetravalent vertex‐transitive graphs of order 4p

A graph is vertex‐transitive if its automorphism group acts transitively on vertices of the graph. A vertex‐transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this article, the tetravalent vertex‐transitive non‐Cayley graphs of order 4p are classified for each prime p. As a result, there are one sporadic and five infinite families of such graphs, of which the sporadic one has order 20, and one infinite family exists for every prime p>3, two families exist if and only if p≡1 (mod 8) and the other two families exist if and only if p≡1 (mod 4). For each family there is a unique graph for a given order. © 2011 Wiley Periodicals, Inc.     

I‐graphs and the corresponding configurations

We consider the class of I‐graphs I(n,j,k), which is a generalization over the class of the generalized Petersen graphs. We study different properties of I‐graphs, such as connectedness, girth, and whether they are bipartite or vertex‐transitive. We give an efficient test for isomorphism of I‐graphs and characterize the automorphism groups of I‐graphs. Regular bipartite graphs with girth at least 6 can be considered as Levi graphs of some symmetric combinatorial configurations. We consider configurations that arise from bipartite I‐graphs. Some of them can be realized in the plane as cyclic astral configurations, i.e., as geometric configurations with maximal isometric symmetry. © 2005 Wiley Periodicals, Inc.     

Vertex Cuts

Given a connected graph, in many cases it is possible to construct a structure tree that provides information about the ends of the graph or its connectivity. For example Stallings' theorem on the structure of groups with more than one end can be proved by analyzing the action of the group on a structure tree and Tutte used a structure tree to investigate finite 2‐connected graphs, that are not 3‐connected. Most of these structure tree theories have been based on edge cuts, which are components of the graph obtained by removing finitely many edges. A new axiomatic theory is described here using vertex cuts, components of the graph obtained by removing finitely many vertices. This generalizes Tutte's decomposition of 2‐connected graphs to k‐connected graphs for any k, in finite and infinite graphs. The theory can be applied to nonlocally finite graphs with more than one vertex end, i.e. ends that can be separated by removing a finite number of vertices. This gives a decomposition for a group acting on such a graph, generalizing Stallings' theorem. Further applications include the classification of distance transitive graphs and k‐CS‐transitive graphs.     

A Class of Bipartite 2-Path-Transitive Graphs

A graph Γ is called half-arc-transitive if it’s automorphism group Aut Γ is transitive on the vertex set and edge set, but not on the arc set of the graph Γ, and it is called 2-path-transitive if Aut Γ is transitive on the set of the 2-paths. In this paper we construct a class of 2-path-transitive graphs from some symmetric groups, based on which a new class of half-arc-transitive graphs is given.     

Cores of Imprimitive Symmetric Graphs of Order a Product of Two Distinct Primes

A retract of a graph Γ is an induced subgraph Ψ of Γ such that there exists a homomorphism from Γ to Ψ whose restriction to Ψ is the identity map. A graph is a core if it has no nontrivial retracts. In general, the minimal retracts of a graph are cores and are unique up to isomorphism; they are called the core of the graph. A graph Γ is G‐symmetric if G is a subgroup of the automorphism group of Γ that is transitive on the vertex set and also transitive on the set of ordered pairs of adjacent vertices. If in addition the vertex set of Γ admits a nontrivial partition that is preserved by G, then Γ is an imprimitive G‐symmetric graph. In this paper cores of imprimitive symmetric graphs Γ of order a product of two distinct primes are studied. In many cases the core of Γ is determined completely. In other cases it is proved that either Γ is a core or its core is isomorphic to one of two graphs, and conditions on when each of these possibilities occurs is given.     

Perfect matchings after vertex deletions

This paper considers some classes of graphs which are easily seen to have many perfect matchings. Such graphs can be considered robust with respect to the property of having a perfect matching if under vertex deletions (with some mild restrictions), the resulting subgraph continues to have a perfect matching. It is clear that you can destroy the property of having a perfect matching by deleting an odd number of vertices, by upsetting a bipartition or by deleting enough vertices to create an odd component. One class of graphs we consider is the m×m lattice graph (or grid graph) for m even. Matchings in such grid graphs correspond to coverings of an m×m checkerboard by dominoes. If in addition to the easy conditions above, we require that the deleted vertices be apart, the resulting graph has a perfect matching. The second class of graphs we consider is a k-fold product graph consisting of k copies of a given graph G with the ith copy joined to the i+1st copy by a perfect matching joining copies of the same vertex. We show that, apart from some easy restrictions, we can delete any vertices from the kth copy of G and find a perfect matching in the product graph with k suitably large.     

Super-connected edge transitive graphs

A graph G is said to be super-connected if any minimum cut of G isolates a vertex. In a previous work due to the second author of this note, super-connected graphs which are both vertex transitive and edge transitive are characterized. In this note, we generalize the characterization to edge transitive graphs which are not necessarily vertex transitive, showing that the only irreducible edge transitive graphs which are not super-connected are the cycles C_n(n?6) and the line graph of the 3-cube, where irreducible means the graph has no vertices with the same neighbor set. Furthermore, we give some sufficient conditions for reducible edge transitive graphs to be super-connected.     

On cubic non‐Cayley vertex‐transitive graphs

In 1983, the second author [D. Maru?i?, Ars Combinatoria 16B (1983), 297–302] asked for which positive integers n there exists a non‐Cayley vertex‐transitive graph on n vertices. (The term non‐Cayley numbers has later been given to such integers.) Motivated by this problem, Feng [Discrete Math 248 (2002), 265–269] asked to determine the smallest valency ?(n) among valencies of non‐Cayley vertex‐transitive graphs of order n. As cycles are clearly Cayley graphs, ?(n)?3 for any non‐Cayley number n. In this paper a goal is set to determine those non‐Cayley numbers n for which ?(n) = 3, and among the latter to determine those for which the generalized Petersen graphs are the only non‐Cayley vertex‐transitive graphs of order n. It is known that for a prime p every vertex‐transitive graph of order p, p² or p³ is a Cayley graph, and that, with the exception of the Coxeter graph, every cubic non‐Cayley vertex‐transitive graph of order 2p, 4p or 2p² is a generalized Petersen graph. In this paper the next natural step is taken by proving that every cubic non‐Cayley vertex‐transitive graph of order 4p², p>7 a prime, is a generalized Petersen graph. In addition, cubic non‐Cayley vertex‐transitive graphs of order 2p^k, where p>7 is a prime and k?p, are characterized. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 77–95, 2012     

3‐Factor‐Criticality of Vertex‐Transitive Graphs

A graph of order n is p ‐factor‐critical, where p is an integer of the same parity as n, if the removal of any set of p vertices results in a graph with a perfect matching. 1‐factor‐critical graphs and 2‐factor‐critical graphs are factor‐critical graphs and bicritical graphs, respectively. It is well known that every connected vertex‐transitive graph of odd order is factor‐critical and every connected nonbipartite vertex‐transitive graph of even order is bicritical. In this article, we show that a simple connected vertex‐transitive graph of odd order at least five is 3‐factor‐critical if and only if it is not a cycle.     

Looseness of Plane Graphs

A face of a vertex coloured plane graph is called loose if the number of colours used on its vertices is at least three. The looseness of a plane graph G is the minimum k such that any surjective k-colouring involves a loose face. In this paper we prove that the looseness of a connected plane graph G equals the maximum number of vertex disjoint cycles in the dual graph G* increased by 2. We also show upper bounds on the looseness of graphs based on the number of vertices, the edge connectivity, and the girth of the dual graphs. These bounds improve the result of Negami for the looseness of plane triangulations. We also present infinite classes of graphs where the equalities are attained.     

Arc-transitive distance-regular covers of cliques with <Emphasis Type="Italic">λ</Emphasis> = µ

We study antipodal distance-regular graphs of diameter 3 such that their automorphism group acts transitively on the set of pairs (a, b), where {a, b} is an edge of the graph. Since the automorphism group of such graphs acts 2-transitively on the set of antipodal classes, the classification of 2-transitive permutation groups can be used. We classify arc-transitive distance-regular graphs of diameter 3 in which any two vertices at distance at most two have exactly µ common neighbors.     

Unitary Graphs

Unitary graphs are arc‐transitive graphs with vertices the flags of Hermitian unitals and edges defined by certain elements of the underlying finite fields. They played a significant role in a recent classification of a class of arc‐transitive graphs that admit an automorphism group acting imprimitively on the vertices. In this article, we prove that all unitary graphs are connected of diameter two and girth three. Based on this, we obtain, for any prime power , a lower bound of order on the maximum number of vertices in an arc‐transitive graph of degree and diameter two.     

Paths with restricted degrees of their vertices in planar graphs

In this paper it is proved that every 3-connected planar graph contains a path on 3 vertices each of which is of degree at most 15 and a path on 4 vertices each of which has degree at most 23. Analogous results are stated for 3-connected planar graphs of minimum degree 4 and 5. Moreover, for every pair of integers n 3, k 4 there is a 2-connected planar graph such that every path on n vertices in it has a vertex of degree k.     

Outerplanar and Planar Oriented Cliques

The clique number of an undirected graph G is the maximum order of a complete subgraph of G and is a well‐known lower bound for the chromatic number of G. Every proper k‐coloring of G may be viewed as a homomorphism (an edge‐preserving vertex mapping) of G to the complete graph of order k. By considering homomorphisms of oriented graphs (digraphs without cycles of length at most 2), we get a natural notion of (oriented) colorings and oriented chromatic number of oriented graphs. An oriented clique is then an oriented graph whose number of vertices and oriented chromatic number coincide. However, the structure of oriented cliques is much less understood than in the undirected case. In this article, we study the structure of outerplanar and planar oriented cliques. We first provide a list of 11 graphs and prove that an outerplanar graph can be oriented as an oriented clique if and only if it contains one of these graphs as a spanning subgraph. Klostermeyer and MacGillivray conjectured that the order of a planar oriented clique is at most 15, which was later proved by Sen. We show that any planar oriented clique on 15 vertices must contain a particular oriented graph as a spanning subgraph, thus reproving the above conjecture. We also provide tight upper bounds for the order of planar oriented cliques of girth k for all .