Related papers: Multiple Kronecker Covering Graphs

A connected $k$-chromatic graph $G$ is double-critical if for all edges $uv$ of $G$ the graph $G - u - v$ is $(k-2)$-colourable. The only known double-critical $k$-chromatic graph is the complete $k$-graph $K_k$. The conjecture that there…

We show by example that there is a Cayley graph, having two invariant random subgraphs X and Y, such that there exists a monotone coupling between them in the sense that $X\subset Y$, although no such coupling can be invariant. Here,…

For a fixed graph $H$, what is the smallest number of colours $C$ such that there is a proper edge-colouring of the complete graph $K_n$ with $C$ colours containing no two vertex-disjoint colour-isomorphic copies, or repeats, of $H$? We…

Let $G=(V,E)$ be a graph of order $n$ and let $1\leq k< n$ be an integer. The $k$-token graph of $G$ is the graph whose vertices are all the $k$-subsets of $V$, two of which are adjacent whenever their symmetric difference is a pair of…

In this paper we study underlying graphs corresponding to a set of halving lines. We establish many properties of such graphs. In addition, we tighten the upper bound for the number of halving lines.

In this article, we give a positive answer to the cycle double cover conjecture. Ones who are mainly interesting in the proof of the conjecture can only read Sections 2 and 4.

We approach the cycle double cover conjecture by looking for a circular 2-cell embedding of cubic graphs on an arbitrary surface. It is easy to see that if such an embedding exists, we can get to it from an arbitrary starting 2-cell…

A certain signed adjacency matrix of the hypercube, which Hao Huang used last year to resolve the sensitivity conjecture, is closely related to the unique, 4-cycle free, 2-fold cover of the hypercube. We develop a framework in which this…

A simple topological graph is $k$-quasiplanar ($k\geq 2$) if it contains no $k$ pairwise crossing edges, and $k$-planar if no edge is crossed more than $k$ times. In this paper, we explore the relationship between $k$-planarity and…

A graph $G$ is called well-covered if all maximal independent sets of vertices have the same cardinality. A well-covered graph $G$ is called uniformly well-covered if there is a partition of the set of vertices of $G$ such that each maximal…

Whitney's Theorem states that every graph, different from $K_3$ or $K_{1,3}$, is uniquely determined by its line graph. A $1$-line graph of a multi-graph is the graph with as vertices the edges of the multi-graph, and two edges adjacent if…

Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimise or maximise the number of connected induced subgraphs. For each of these classes, we find that the graphs that minimise the number of…

We prove that any graph of multicurves satisfying certain natural properties is either hyperbolic, relatively hyperbolic, or thick. Further, this geometric characterization is determined by the set of subsurfaces that intersect every vertex…

We consider the problem of covering an input graph $H$ with graphs from a fixed covering class $G$. The classical covering number of $H$ with respect to $G$ is the minimum number of graphs from $G$ needed to cover the edges of $H$ without…

We extend the closed graph theorem and the open mapping theorem to a context in which a natural duality interchanges their extensions.

We determine to within a constant factor the threshold for the property that two random k-uniform hypergraphs with edge probability p have an edge-disjoint packing into the same vertex set. More generally, we allow the hypergraphs to have…

We recall several known results about minimally 2-connected graphs, and show that they all follow from a decomposition theorem. Starting from an analogy with critically 2-connected graphs, we give structural characterizations of the classes…

A linear layout of a graph typically consists of a total vertex order, and a partition of the edges into sets of either non-crossing edges, called stacks, or non-nested edges, called queues. The stack (queue) number of a graph is the…

Graphs are ubiquitous in nature and can therefore serve as models for many practical but also theoretical problems. For this purpose, they can be defined as many different types which suitably reflect the individual contexts of the…

A simple topological graph $G$ is a graph drawn in the plane so that any pair of edges have at most one point in common, which is either an endpoint or a proper crossing. $G$ is called saturated if no further edge can be added without…