Abstract: Using covering graph techniques, a structural result about connected cubic simple graphs admitting an edge-transitive solvable group of automorphisms is proved. This implies, among other, that every such graph can be obtained from either the 3-dipole Dip3 or the complete graph K4, by a sequence of elementary-abelian covers. Another consequence of the main structural result is that the action of an arc-transitive solvable group on a connected cubic simple graph is at most 3-arc-transitive. As an application, a new infinite family of semisymmetric cubic graphs, arising as regular elementary abelian covering projections of K3,3, is constructed.

Abstract: We consider the Goldberg-Coxeter construction $GC_{k,l}(G_0)$ (a generalization of a simplicial subdivision of the dodecahedron considered by Goldberg [Tohoku Mathematical Journal, 43 (1937) 104–108] and Coxeter [A Spectrum of Mathematics, OUP, (1971) 98–107]), which produces a plane graph from any $3$- or $4$-valent plane graph for integer parameters $k,l$. A zigzag in a plane graph is a circuit of edges, such that any two, but no three, consecutive edges belong to the same face; a central circuit in a $4$-valent plane graph $G$ is a circuit of edges, such that no two consecutive edges belong to the same face. We study the zigzag (or central circuit) structure of the resulting graph using the algebraic formalism of the moving group , the $(k,l)$-product and a finite index subgroup of $SL_2(\Bbb{Z})$, whose elements preserve the above structure. We also study the intersection pattern of zigzags (or central circuits) of $GC_{k,l}(G_0)$ and consider its projections , obtained by removing all but one zigzags (or central circuits).

Abstract: In this note we show how 1-factors in the middle two layers of the discrete cube can be used to construct 2-factors in the Odd graph (the Kneser graph of (k − 1)-sets from a (2k − 1)-set). In particular, we use the lexical matchings of Kierstead and Trotter, and the modular matchings of Duffus, Kierstead and Snevily, to give explicit constructions of two different 2-factorisations of the Odd graph.

Abstract: We prove that in a certain statistical sense the Cayley graph of almost every finitely presented group with $m\ge 2$ generators contains a subdivision of the complete graph on $l\le 2m+1$ vertices. In particular, this Cayley graph is non planar. We also show that some group constructions preserve the planarity.

Abstract: We propose a new elementary definition of the Higman-Sims graph in which the 100 vertices are parametrised with ${\Bbb Z}_4\times{\Bbb Z}_5\times{\Bbb Z}_5$ and adjacencies are described by linear and quadratic equations. This definition extends Robertson's pentagon-pentagram definition of the Hoffman-Singleton graph and is obtained by studying maximum cocliques of the Hoffman-Singleton graph in Robertson's parametrisation. The new description is used to count the 704 Hoffman-Singleton subgraphs in the Higman-Sims graph, and to describe the two orbits of the simple group HS on them, including a description of the doubly transitive action of HS within the Higman-Sims graph. Numerous geometric connections are pointed out. As a by-product we also have a new construction of the Steiner system $S(3,6,22)$.

Abstract: We construct a new symmetric Hamilton cycle decomposition of the complete graph Kn for odd n > 7. © 2003 Wiley Periodicals, Inc.

Abstract: We show that in any graph G on n vertices with d(x) + d(y) ≥ n for any two nonadjacent vertices x and y, we can fix the order of k vertices on a given cycle and find a hamiltonian cycle encountering these vertices in the same order, as long as k < n/12 and G is d(k + 1)/2e-connected. Further we show that every b3k/2cconnected graph on n vertices with d(x) + d(y) ≥ n for any two nonadjacent vertices x and y is k-ordered hamiltonian, i.e. for every ordered set of k vertices we can find a hamiltonian cycle encountering these vertices in the given order. Both connectivity bounds are best possible.

Abstract: In this paper, we examine a method for the fusion of ranked data in the context of a Cayley graph. We investigate this Cayley graph model for optimization of fusion by rank combination. We outline a method of data fusion by combination of weighted rankings. Information systems are represented as nodes in a Cayley graph. Our goal is to determine a metric of diversity and performance in this graph in order to build a model for optimizing fusion by rank combination. We use the Kendall distance between nodes in the Cayley graph of the symmetric group Sn as a measure of performance. In doing so we demonstrate that in S6 there is a quadratic relationship between the weights of the fusion of two information systems and the performance of the fusion in our abstract space. From such a relationship we propose a set of functions for extrapolating optimal fusion weights in the symmetric group Sn.

Abstract: The problem of colouring a k -colourable graph is well-known to be NP-complete, for k ≥ 3. The MAX- k -CUT approach to approximate k -colouring is to assign k colours to all of the vertices in polynomial time such that the fraction of 'defect edges' (with endpoints of the same colour) is provably small. The best known approximation was obtained by Frieze and Jerrum (1997), using a semidefinite programming (SDP) relaxation which is related to the Lovasz ϑ -function. In a related work, Karger et al. (1998) devised approximation algorithms for colouring k -colourable graphs exactly in polynomial time with as few colours as possible. They also used an SDP relaxation related to the ϑ -function. In this paper we further explore semidefinite programming relaxations where graph colouring is viewed as a satisfiability problem, as considered in De Klerk et al. (2000). We first show that the approximation to the chromatic number suggested in De Klerk et al. (2000) is bounded from above by the Lovasz ϑ -function. The underlying semidefinite programming relaxation in De Klerk et al. (2000) involves a lifting of the approximation space, which in turn suggests a provably good MAX- k -CUT algorithm. We show that of our algorithm is closely related to that of Frieze and Jerrum; thus we can sharpen their approximation guarantees for MAX- k -CUT for small fixed values of k .F or example, if k = 3w e can improve their bound from 0 . 832718 to 0 . 836008, and for k = 4 from 0 . 850301 to 0 . 857487. We also give a new asymptotic analysis of the Frieze-Jerrum rounding scheme, that provides a unifying proof of the main results of both Frieze and Jerrum (1997) and Karger et al. (1998) for k � 0.

Abstract: Hajos theorem states that every graph with chromatic number at least k can be obtained from the complete graph K k by a sequence of simple operations such that every intermediate graph also has chromatic number at least k. Here, Hajos theorem is extended in three slightly different ways to colorings and circular colorings of edge-weighted graphs. These extensions shed some new light on the Hajos theorem and show that colorings of edge-weighted graphs are most natural extension of usual graph colorings.

Abstract: We develop a combinatorial method to show that the dodecahedron graph has, up to rotation and reflection, a unique Hamiltonian cycle. Platonic graphs with this property are called topologically uniquely Hamiltonian. The same method is used to demonstrate topologically distinct Hamiltonian cycles on the icosahedron graph and to show that a regular graph embeddable on the 2-holed torus is topologically uniquely Hamiltonian.

Abstract: Suppose G is a series-parallel graph. It was proved in [3] that either χc(G) = 3 or χc(G) ≤ 8-3. So none of the rationals in the interval (8-3, 3) is the circular chromatic number of a series-parallel graph. This paper proves that for every rational re [2, 8-3] ∪{3} there exists a series-parallel graph G with χc(G) = r. © 2004 Wiley Periodicals, Inc. J Graph Theory 46:5768, 2004

Abstract: The Kneser graph K(n, k) is the graph whose vertices are the k-element subsets of an n-element set, with two vertices adjacent if the sets are disjoint. The chromatic number of the Kneser graph K(n, k) is n−2k+2. Zoltan Furedi raised the question of determining the chromatic number of the square of the Kneser graph, where the square of a graph is the graph obtained by adding edges joining vertices at distance at most 2. We prove that χ(K2(2k+1, k))≤4k when k is odd and χ(K2(2k+1, k))≤4k+2 when k is even. Also, we use intersecting families of sets to prove lower bounds on χ(K2(2k+1, k)), and we find the exact maximum size of an intersecting family of 4-sets in a 9-element set such that no two members of the family share three elements.

Abstract: An undirected graph is said to be edge-regular with parameters if it has vertices, each vertex has degree , and each edge belongs to triangles. We put . Brouwer, Cohen, and Neumaier proved that every connected edge-regular graph with (equivalently, with ) is strongly regular. In this paper we construct an example of an edge-regular, not strongly regular graph on 36 vertices with . This shows that the estimate above is sharp. We prove that every connected edge-regular graph with (equivalently, either satisfies , or has parameters or , or is strongly regular.

Abstract: In this paper, we prove that a cubic line graph G on n vertices rather than the complete graph K4 has * vertex-disjoint triangles and the vertex independence number *. Moreover, the equitable chromatic number, acyclic chromatic number and bipartite density of G are * respectively.

Abstract: Generalized Petersen graphs iare an important class of commonly used in- terconnection networks and have been studied by various researchers. In this paper, we show that the diameter of generalized Petersen graph P(m, 2) is O(m/4) and the 3-wide diameter of P(m, 2) is O(m/3).

Abstract: We provide the list of all paths with at most 16 arcs with the property that if a graph G admits an orientation G such that one of the paths in our list admits no homomorphism to G, then G is 3-colourable.

Abstract: The group of automorphisms of the 32-vertex Dyck graph is identified as the tetrakisoctahedral group, 4 O. This group has 96 elements and conserves orientation on the standard embedding of the Dyck graph on a surface of genus 3, consisting of 12 octagons. An alternative regular map of the Dyck graph on a torus is found, which is made up of 16 hexagons. Orientation on this surface is conserved by another group of 96 elements, 4 Th , which is non-isomorphic to 4 O. The subgroup structures of 4 O and 4 Th are derived, and character tables of 4 O and some of its subgroups are constructed. The symmetry representations of the Dyck graph and its topological dual are determined. Finally a molecular realization of the Dyck graph on the genus-3 ‘Plumber's nightmare’ is proposed, which can be considered as a new type of octagonal carbon network.

Abstract: We consider the lossless compression of vertex transitive graphs. An undirected graph G = (V, E) is called vertex transitive if for every pair of vertices x, y ∈ V , there is an automorphism σ of G, such that σ(x) = y. A result due to Sabidussi, guarantees that for every vertex transitive graph G there exists a graph mG (m is a positive integer) which is a Cayley graph. We propose as the compressed form of G a finite presentation (X, R) , where (X, R) presents the group Γ corresponding to such a Cayley graph mG. On a conjecture, we demonstrate that for a large subfamily of vertex transitive graphs, the original graph G can be completely reconstructed from its compressed representation.