Automorphisms of graphs and coverings

We apply the theory of covering spaces to show how one can construct infinitely many finite s-transitive or locally s-transitive graphs. N. Biggs has used for similar purpose a special graph covering construction due to J. H. Conway.     

Projective-planar double coverings of graphs

We shall show that a connected graph G is projective-planar if and only if it has a projective-planar double covering and that any projective-planar double covering of a 2-connected nonplanar graph is planar.     

On the coverings of graphs

Let ρ(n) denote the smallest integer with the property that any graph with n vertices can be covered by ρ(n) complete bipartite subgraphs. We prove a conjecture of J.-C. Bermond by showing $\text{ρ(n)=n+o(n}{}^{\mathrm{<mtext>11</mtext><mtext>14+?</mtext>}}\text{)}$ for any positive ?.     

Extremal clique coverings of complementary graphs

Letcc(G) (resp. cp(G)) be the least number of complete subgraphs needed to cover (resp. partition) the edges of a graphG. We present bounds on max {cc(G)+cc(Ḡ)}, max {cp(G)+cp(Ḡ)}, max {cc(G)cc(Ḡ)} and max {cp(G)cp(Ḡ)} where the maxima are taken over all graphsG onn vertices and Ḡ is the complement ofG inK _n . Several related open problems are also given.     

Cycle and cocycle coverings of graphs

In this article, we show that for any simple, bridgeless graph G on n vertices, there is a family ?? of at most n?1 cycles which cover the edges of G at least twice. A similar, dual result is also proven for cocycles namely: for any loopless graph G on n vertices and ε edges having cogirth g^*?3 and k(G) components, there is a family of at most ε?n+k(G) cocycles which cover the edges of G at least twice. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 270–284, 2010     

Clique coverings and partitions of line graphs

A clique in a graph G is a complete subgraph of G. A clique covering (partition) of G is a collection C of cliques such that each edge of G occurs in at least (exactly) one clique in C. The clique covering (partition) numbercc(G) (cp(G)) of G is the minimum size of a clique covering (partition) of G. This paper gives alternative proofs, using a unified approach, for the results on the clique covering (partition) numbers of line graphs obtained by McGuinness and Rees [On the number of distinct minimal clique partitions and clique covers of a line graph, Discrete Math. 83 (1990) 49-62]. We also employ the proof techniques to give an alternative proof for the De Brujin-Erd?s Theorem.     

A note on coverings of plane graphs

For any plane graph G the number of edges in a minimum edge covering of the faces of G is at most the vertex independence number of G and the numbre of vertices in a minimum vertex covering of the faces of G is at most the edge independence number of G. © 1995 John Wiley & Sons, Inc.     

Scattering matrices of regular coverings of graphs

We give a decomposition formula for the determinant on the bond scattering matrix of a regular covering of G. Furthermore, we define an L-function of G, and give a determinant expression of it. As a corollary, we express the determinant on the bond scattering matrix of a regular covering of G by means of its L-functions.     

Universal coverings of PL-manifolds via coloured graphs

Summary By means of techniques and results concerning maps on surfaces [JS] and edge-coloured graphs representing PL-manifolds [FGG], we prove the existence of an infinite ball complexP(n), n > 1, such thatevery orientable PL-manifold of dimension n is a quotient of |P(n)| by the action of a finite index subgroup of a Fuchsian group with signature ,with h(2) = h(3) = 4 and h(n) = 2, for n > 3. The core of the proof is that all orientable PL-manifolds of dimensionn can be represented by edge-coloured graphs which are quotients of a universal graph, only depending onn.     

Components of branched coverings of current graphs

Desired graph imbeddings are obtained as branched coverings of simpler imbeddings, continuing earlier work of the authors. This paper inaugurates a systematic approach to the problem of devising appropriate current assignments. Chief results include the establishment of a bijective correspondence between the components of the cover and the cosets of the isotropy group of an arbitrary component, leading to a proof that the components are mutually isomorphic regular branched coverings of the dual of the current graph imbedding. Also given is a characterization of the isotropy group via a face labeling technique that aids in the construction of imbeddings.     

Total matchings and total coverings of graphs

In graph theory, the related problems of deciding when a set of vertices or a set of edges constitutes a maximum matching or a minimum covering have been extensively studied. In this paper we generalize these ideas by defining total matchings and total coverings, and show that these sets, whose elements in general consist of both vertices and edges, provide a way to unify these concepts. Parameters denoting the maximum and the minimum cardinality of these sets are introduced and upper and lower bounds depending only on the order of the graph are obtained for the number of elements in arbitrary total matchings and total coverings. Precise values of all the parameters are found for several general classes of graphs, and these are used to establish the sharpness of most of the bounds. In addition, variations of some well known equalities due to Gallai relating covering and matching numbers are obtained.     

A note on line coverings of graphs

Let minimum, maximum, refer to the number of elements in a set and let minimal, maximal refer to set inclusion. It is shown that a minimal cover of a graph is minimum if and only if it contains a maximum matching of that graph; a maximal matching of a graph is maximum if and only if it is contained in a minimum cover of that graph diminished by the set of its isolated points.     

Vertex coverings by monochromatic paths and cycles

We survey some results on covering the vertices of 2-colored complete graphs by two paths or by two cycles of different color. We show the role of these results in determining path Ramsey numbers and in algorithms for finding long monochromatic paths or cycles in 2-colored complete graphs.     

Weighted scattering matrices of regular coverings of graphs

We give a decomposition formula for the determinant det(I ? U(λ)) of the weighted bond scattering matrix U(λ) of a regular covering of G. Furthermore, we define an L-function of G, and give a determinant expression of it. As a corollary, we express some determinant of the weighted bond scattering matrix of a regular covering of G by means of its L-functions.     

On factor-invariant graphs with two cycles

We classify trivalent vertex-transitive graphs whose edge sets have a partition into a 2-factor composed of two cycles and a 1-factor that is invariant under the action of the automorphism group.     

Edge-colored complete graphs with alternating cycles

We prove that if the edges of the complete graph on n≥6 vertices are colored so that no vertex is on more than Δ edges of the same color, 1<Δ<n?2, then the graph has cycles of all lengths 3 through n with no Δ consecutive edges the same color.