Related papers: Cantor's Powerset Theorem, Graph-Theoretically
Cantor sets of integers have a rich set of arithmetic combinatorial properties. We consider classical Cantor sets, with a base and a fixed set of allowed digits. For such sets, we (a) give examples of such sets that satisfy the intersective…
We prove that Menger's theorem is valid for infinite graphs, in the following strong form: let $A$ and $B$ be two sets of vertices in a possibly infinite digraph. Then there exist a set $\cp$ of disjoint $A$-$B$ paths, and a set $S$ of…
We generalize the enhanced power graph by replacing elements with classes under automorphisms. We show that the connectivity and diameter of this graph is similar to that of the enhanced power graph. We consider the universal vertices of…
We discuss counting problems linked to finite versions of Cantor's diagonal of infinite tableaux. We extend previous results of [2] by refining an equivalence relation that reduces significantly the exhaustive generation. New enumerative…
In this note, we present a simple directed graph proof of Sharkovsky's theorem.
We offer a new perspective on the closed graph theorem and the open mapping theorem for separated barrelled spaces and fully complete spaces.
The study of power domination in graphs arises from the problem of placing a minimum number of measurement devices in an electrical network while monitoring the entire network. A power dominating set of a graph is a set of vertices from…
We give an elementary, self-contained, and purely combinatorial proof of the Rayleigh monotonicity property of graphs.
In this paper I survey the sources of inspiration for my own and co-authored work in trying to develop a general theory of graph polynomials. I concentrate on meta-theorems, i.e., theorem which depend only on the form infinite classes of…
A graph theoretic perspective is taken for a range of phenomena in continuum physics in order to develop representations for analysis of large scale, high-fidelity solutions to these problems. Of interest are phenomena described by partial…
The purpose of this note is to prove that the set of connective constants of Cayley graphs contains a Cantor space.
We study some versions of the statement of Hadwiger's conjecture for finite as well as infinite graphs.
This paper examines the possibilities of extending Cantor's two arguments on the uncountable nature of the set of real numbers to one of its proper denumerable subsets: the set of rational numbers. The paper proves that, unless certain…
We present a conceptually new approach to describe state-of-the-art photonic quantum experiments using Graph Theory. There, the quantum states are given by the coherent superpositions of perfect matchings. The crucial observation is that…
In this paper we discuss reconstruction problems for graphs. We develop some new ideas like isomorphic extension of isomorphic graphs, partitioning of vertex sets into sets of equivalent points, subdeck property, etc. and develop an…
In a 1977 paper, Steffens identified an elegant criterion for determining when a countable graph has a perfect matching. In this paper, we will investigate the proof-theoretic strength of this result and related theorems. We show that a…
We give a constructive proof of Carpenter's Theorem due to Kadison. Unlike the original proof our approach also yields the real case of this theorem.
In this note we prove that the vertex energy of a graph, as defined in Arizmendi and Juarez (2018), can be calculated in terms of a Coulson integral formula. We present examples of how this formula can be used, and we show some applications…
Working in any model theoretic structure, we single out a class of definable bipartite graphs that admit definable, close to perfect matchings. We use this result to prove a strengthening of Tarski's theorem for the definable setting.
We study topological properties of the graph topology.