Related papers: A note on universal graphs for spanning trees
By the theorem of Mantel $[5]$ it is known that a graph with $n$ vertices and $\lfloor \frac{n^{2}}{4} \rfloor+1$ edges must contain a triangle. A theorem of Erd\H{o}s gives a strengthening: there are not only one, but at least…
The \emph{spanning tree packing number} of a graph $G$ is the maximum number of edge-disjoint spanning trees contained in $G$. Let $k\geq 1$ be a fixed integer. Palmer and Spencer proved that in almost every random graph process, the…
How many edges in an $n$-vertex graph will force the existence of a cycle with as many chords as it has vertices? Almost 30 years ago, Chen, Erd\H{o}s and Staton considered this question and showed that any $n$-vertex graph with $2n^{3/2}$…
A graph $G$ is $\textit{universal}$ for a (finite) family $\mathcal{H}$ of graphs if every $H \in \mathcal{H}$ is a subgraph of $G$. For a given family $\mathcal{H}$, the goal is to determine the smallest number of edges an…
It is well known that finding extremal values and structures can be hard in weighted graphs. However, if the weights are random, this problem can become way easier. In this paper, we examine the minimal weight of a union of $k$…
While the notion of arboricity of a graph is well-known in graph theory, very few results are dedicated to the minimal number of trees covering the edges of a graph, called the tree number of a graph.
The paper shows that almost every $n$-vertex graph is such that the multiset of its induced subgraphs on $3 \log_2{n}$ vertices is sufficient to determine it up to isomorphism. Therefore, for checking the isomorphism of a pair of $n$-vertex…
We introduce the inverse Voronoi diagram problem in graphs: given a graph $G$ with positive edge-lengths and a collection $\mathbb{U}$ of subsets of vertices of $V(G)$, decide whether $\mathbb{U}$ is a Voronoi diagram in $G$ with respect to…
We prove that every graph $G$ on $n$ vertices with no isolated vertices contains an induced subgraph of size at least $n/10000$ with all degrees odd. This solves an old and well-known conjecture in graph theory.
This is an expository paper. A $1$-cycle in a graph is a set $C$ of edges such that every vertex is contained in an even number of edges from $C$. E.g., a cycle in the sense of graph theory is a $1$-cycle, but not vice versa. It is easy to…
We obtain sharp bounds for the number of n-cycles in a finite graph as a function of the number of edges, and prove that the complete graph is optimal in more ways than could be imagined. En route, we prove some sharp estimates on power…
We show that the problem of the existence of universal graphs with specified forbidden subgraphs can be systematically reduced to certain critical cases by a simple pruning technique which simplifies the underlying structure of the…
We study the problem of connecting the parts of a multipartite graph using a minimum number of edges under a matching constraint. We introduce interconnection trees, defined as matchings whose projections onto the quotient graph form a…
The subpath number of a graph G is defined as the total number of subpaths in G, and it is closely related to the number of subtrees, a well-studied topic in graph theory. This paper is a continuation of our previous paper [5], where we…
The degree sequence of a graph is a numerical method to characterize the properties of graphs. Generalized forms of degree sequences exist for complete graphs and complete graphs. Nikolopolus et al. characterized the number of spanning…
The segment number of a planar graph $G$ is the smallest number of line segments needed for a planar straight-line drawing of $G$. Dujmovi\'c, Eppstein, Suderman, and Wood [CGTA'07] introduced this measure for the visual complexity of…
Let $f(n,k)$ be the minimum number of edges that must be removed from some complete geometric graph $G$ on $n$ points, so that there exists a tree on $k$ vertices that is no longer a planar subgraph of $G$. In this paper we show that…
In this paper we examine the classes of graphs whose $K_n$-complements are trees and quasi-threshold graphs and derive formulas for their number of spanning trees; for a subgraph $H$ of $K_n$, the $K_n$-complement of $H$ is the graph…
An added edge to a graph is called an inset edge. Predicting k inset edges which minimize the average distance of a graph is known to be NP-Hard. However, when k = 1 the complexity of the problem is polynomial. In this paper, some tools for…
The treewidth of a graph is an important invariant in structural and algorithmic graph theory. This paper studies the treewidth of line graphs. We show that determining the treewidth of the line graph of a graph $G$ is equivalent to…