Related papers: On the Induced Neighbourhood of Vertex-Transitive …
We consider the problem of covering a graph with a given number of induced subgraphs so that the maximum number of vertices in each subgraph is minimized. We prove NP-completeness of the problem, prove lower bounds, and give approximation…
We give two new conditions on topological $k$-graphs that are equivalent to the Yeend's aperiodicity Condition (A). Each of the new conditions concerns finite paths rather than infinite. We use a specific example, resulting from a new…
Graphs constructed to translate some graph problem into another graph problem are usually called auxiliary graphs. Specifically total graphs of simple graphs are used to translate the total colouring problem of the original graph into a…
In this paper, we introduce a generalized concept of vertex transitivity in graphs called generalized vertex transitivity. We put forward a new invariant called transitivity number of a graph. The value of this invariant in different…
We prove that a graph $G$ contains no induced $5$-vertex path and no induced complement of a $5$-vertex path if and only if $G$ is obtained from $5$-cycles and split graphs by repeatedly applying the following operations: substitution,…
One must add arrows which are forced by transitivity to form the transitive closure of a directed graph. We introduce a construction of a transitive directed graph which is formed by adding vertices instead of arrows and which preserves the…
We prove new lower bounds on the crossing number of a complete graphs assuming that it is drawn in such a way that it contains a Hamiltonian cycle with no crossings.
We study transitivity properties of graphs with more than one end. We completely classify the distance-transitive such graphs and, for all $k \geq 3$, the $k$-CS-transitive such graphs.
It is well-known that in finite graphs, large complete minors/topological minors can be forced by assuming a large average degree. Our aim is to extend this fact to infinite graphs. For this, we generalise the notion of the relative end…
We apply model theoretic methods to the problem of existence of countable universal graphs with finitely many forbidden connected subgraphs. We show that to a large extent the question reduces to one of local finiteness of an…
We prove that for any weakly convergent sequence of finite graphs with bounded vertex degrees, there exists a topological limit graphing.
Let k be a natural number. We introduce k-threshold graphs. We show that there exists an O(n^3) algorithm for the recognition of k-threshold graphs for each natural number k. k-Threshold graphs are characterized by a finite collection of…
For a given finite class of finite graphs H, a graph G is called a realization of H if the neighbourhood of its any vertex induces the subgraph isomorphic to a graph of H. We consider the following problem known as the Generalized…
In a series of papers, of which this is the first, we study sufficient conditions for Hamiltonicity in terms of forbidden induced subgraphs and extend such results to locally finite infinite graphs. For this we use topological circles…
We propose a definition of graph subshifts of finite type that can be seen as extending both the notions of subshifts of finite type from classical symbolic dynamics and finitely presented groups from combinatorial group theory. These are…
In this short note, we give a new sufficient condition for the existence of long cycles in graphs involving Fan-type degree condition and neighborhood intersection.
Developing further Stein's recent notion of relative end degrees in infinite graphs, we investigate which degree assumptions can force a locally finite graph to contain a given finite minor, or a finite subgraph of given minimum degree.…
We give a sufficient condition on totally disconnected topological graphs such that their associated topological graph algebras are purely infinite.
In this paper we extend a result about a sufficient condition for Hamiltonicity for finite graphs by Broersma and Veldmann to locally finite graphs. In order to do this we use topological circles within the Freudenthal compactification of a…
In this paper, we examine the structure of vertex- and edge-transitive strongly regular graphs, using normal quotient reduction. We show that the irreducible graphs in this family have quasiprimitive automorphism groups, and prove (using…