Related papers: The fiber dimension of a graph
We give lower bounds on the size and total size of clique partitions of a graph in terms of its spectral radius and minimum degree, and derive a spectral upper bound for the maximum number of edge-disjoint $t$-cliques. The extremal graphs…
This article provides structural characterization of simple graphs whose edge-set can be partitioned into maximum matchings. We use Vizing's classification of simple graphs based on edge chromatic index.
Given graphs H_1,...,H_k, we study the minimum order of a graph G such that for each i, the induced copies of H_i in G cover V(G). We prove a general upper bound of twice the sum of the numbers m_i, where m_i is one less than the order of…
Graph is an important data representation which appears in a wide diversity of real-world scenarios. Effective graph analytics provides users a deeper understanding of what is behind the data, and thus can benefit a lot of useful…
Let $\mathcal{A}$ be a set of positive numbers. A graph $G$ is called an $\mathcal{A}$-embeddable graph in $\mathbb{R}^d$ if the vertices of $G$ can be positioned in $\mathbb{R}^d$ so that the distance between endpoints of any edge is an…
The concept of metric dimension has applications in a variety of fields, such as chemistry, robotic navigation, and combinatorial optimization. We show bounds for graphs with $n$ vertices and metric dimension $\beta$. For Hamiltonian…
We prove that every graph of rank-width $k$ is a pivot-minor of a graph of tree-width at most $2k$. We also prove that graphs of rank-width at most 1, equivalently distance-hereditary graphs, are exactly vertex-minors of trees, and graphs…
Graphs are a representation of structured data that captures the relationships between sets of objects. With the ubiquity of available network data, there is increasing industrial and academic need to quickly analyze graphs with billions of…
Metric graphs are often introduced based on combinatorics, upon "associating" each edge of a graph with an interval; or else, casually "gluing" a collection of intervals at their endpoints in a network-like fashion. Here we propose an…
We describe the structure of connected graphs with the minimum and maximum average distance, radius, diameter, betweenness centrality, efficiency and resistance distance, given their order and size. We find tight bounds on these graph…
The degree/diameter problem for mixed graphs asks for the largest possible order of a mixed graph with given diameter and degree parameters. Similarly the \emph{degree/geodecity} problem concerns the smallest order of a $k$-geodetic mixed…
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…
Given a graph $G$, an {\em obstacle representation} of $G$ is a set of points in the plane representing the vertices of $G$, together with a set of connected obstacles such that two vertices of $G$ are joined by an edge if and only if the…
We investigate the possibility of proving upper bounds on Hadwiger's number of a graph with partial information, mirroring several known upper bounds for the chromatic number. For each such bound we determine whether the corresponding bound…
The aim of this paper is to generalize the notion of the coloring complex of a graph to hypergraphs. We present three different interpretations of those complexes -- a purely combinatorial one and two geometric ones. It is shown, that most…
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.
Graph drawing research traditionally focuses on producing geometric embeddings of graphs satisfying various aesthetic constraints. After the geometric embedding is specified, there is an additional step that is often overlooked or ignored:…
The maximum number of vertices in a graph of maximum degree $\Delta\ge 3$ and fixed diameter $k\ge 2$ is upper bounded by $(1+o(1))(\Delta-1)^{k}$. If we restrict our graphs to certain classes, better upper bounds are known. For instance,…
In the paper we give a lower bound for the number of vertices of a given graph using its chromatic number. We find the graphs for which this bound is exact. The results are applied in the theory of Foklman numbers.
What is the dimension of a network? Here, we view it as the smallest dimension of Euclidean space into which nodes can be embedded so that pairwise distances accurately reflect the connectivity structure. We show that a recently proposed…