In this paper we introduce some general necessary conditions for the existence of graph homomorphisms, which hold in both directed and undirected cases. Our method is a combination of Diaconis and Saloff– Coste comparison technique for Markov chains and a generalization of Haemers interlacing theorem. As some applications, we obtain a necessary condition for the spanning subgraph problem, which...

A transitive decomposition is a pair (Γ,P) where Γ is a graph and P is a partition of the arc set of Γ such that there is a subgroup of automorphisms of Γ which leaves P invariant and transitively permutes the parts in P. In an earlier paper we gave a characterisation of G-transitive decompositions where Γ is the graph product Km×Km and G is a rank 3 group of product action type. This character...

The Johnson graph J(v, k) has, as vertices, the k-subsets of a v-set V and as edges the pairs of k-subsets with intersection of size k − 1. We introduce the notion of a neighbour-transitive code in J(v, k). This is a vertex subset Γ such that the subgroup G of graph automorphisms leaving Γ invariant is transitive on both the set Γ of ‘codewords’ and also the set of ‘neighbours’ of Γ, which are ...

The expected n-step return-probability EμP [X̂n = o] of a random walk X̂n with symmetric transition probabilities on a random partial graph of a regular graph G of degree δ with transitive automorphism group Aut(G) is considered. The law μ of the random edge-set is assumed to be stationary with respect to some transitive, unimodular subgroup Γ of Aut(G). By the spectral theory of finite random wa...

In distributed networks, a target party T could be a person never meet with a source party S, therefore S may not hold any prior evaluation of trustworthiness of T . To get permit to access S, T should be somewhat trusted by S. Consequently, we should study the approach to evaluate trustworthiness of T . To attack the problem, we view individual participant in distributed networks as a node of ...

let f, g and h be non-empty graphs. the notation f → (g,h) means that if any edge of f is colored by red or blue, then either the red subgraph of f con- tains a graph g or the blue subgraph of f contains a graph h. a graph f (without isolated vertices) is called a ramsey (g,h)−minimal if f → (g,h) and for every e ∈ e(f), (f − e) 9 (g,h). the set of all ramsey (g,h)−minimal graphs is denoted by ...

Cluster Deletion and Cluster Editing ask to transform a graph by at most k edge deletions or edge edits, respectively, into a cluster graph, i.e., disjoint union of cliques. Equivalently, a cluster graph has no conflict triples, i.e., two incident edges without a transitive edge. We solve the two problems in time O∗(1.415k) and O∗(1.76k), respectively. These results round off our earlier work b...

The 2-restricted edge-connectivity λ′′ of a graph G is defined to be the minimum cardinality |S| of a set S of edges such thatG−S is disconnected and is of minimum degree at least two. It is known that λ′′ ≤ g(k − 2) for any connected k-regular graph G of girth g other than K4, K5 and K3,3, where k ≥ 3. In this paper, we prove the following result: For a connected vertex-transitive graph of ord...

This is an annotated bibliography on fully dynamic algorithms for path problems on general directed graphs. In particular, we consider two fundamental problems: dynamic transitive closure and dynamic shortest paths. Although research on these problems spans over more than three decades, in the last couple of years many novel algorithmic techniques have been proposed. 1 Dynamic Path Problems A d...

Graphs possessing a high degree of symmetry have often been considered in topological graph theory. For instance, a number of constructions of genus embeddings by means of current or voltage graphs is based on the observation that a graph can be represented as a Cayley graph for some group. Another kind of embedding problems where symmetrical graphs are encountered is connected with regular map...