Related papers: A Graph Theoretical Approach to the Collatz Proble…

It is well known that the following Collatz Conjecture is one of the unsolved problems in mathematics. Collatz Conjecture: For any positive integer $n>1$, the following recursive algorithm will convergent to 1 by a finite number of steps.…

The famous 3x + 1 problem of L. Collatz needs no introduction; however, this paper concerns a lesser-known, but similarly unresolved, precursor problem : the Original Collatz Conjecture, or OCC. We demonstrate that the core arithmetic…

A graph is chordal if it contains no induced cycle of length four or more. While finite chordal graphs are precisely those admitting tree-decompositions into cliques, this fails for infinite graphs. We establish two results extending the…

Stark and Terras introduced the edge zeta function of a finite graph in 1996. The edge zeta function is the reciprocal of a polynomial in twice as many variables as edges in the graph and can be computed in polynomial time. We look at graph…

The aim of this chapter is to provide an adequate graph theoretic framework for the description of periodic bifurcations which have recently been discovered in descendant trees of finite p-groups. The graph theoretic concepts of rooted…

Tuza (1981) conjectured that the size $\tau(G)$ of a minimum set of edges that intersects every triangle of a graph $G$ is at most twice the size $\nu(G)$ of a maximum set of edge-disjoint triangles of $G$. In this paper we present three…

A linear forest is a forest in which every connected component is a path. The linear arboricity of a graph $G$ is the minimum number of linear forests of $G$ covering all edges. In 1980, Akiyama, Exoo and Harary proposed a conjecture, known…

An order-theoretic forest is a countable partial order such that the set of elements larger than any element is linearly ordered. It is an order-theoretic tree if any two elements have an upper-bound. The order type of a branch can be any…

This paper begins the study of reconfiguration of zero forcing sets, and more specifically, the zero forcing graph. Given a base graph $G$, its zero forcing graph, $\mathscr{Z}(G)$, is the graph whose vertices are the minimum zero forcing…

If $G$ is a strongly connected finite directed graph, the set $\mathcal{T}G$ of rooted directed spanning trees of $G$ is naturally equipped with a structure of directed graph: there is a directed edge from any spanning tree to any other…

A linear forest is a collection of vertex-disjoint paths. The Linear Arboricity Conjecture states that every graph of maximum degree $\Delta$ can be decomposed into at most $\lceil(\Delta+1)/2\rceil$ linear forests. We prove that $\Delta/2…

We prove that for any finite tree $T$ with $n$ vertices and maximal degree $3$, there is a topological embedding of $T$ into the integer grid $Z^2$ which maps vertices to vertices and whose image meets at most $\frac{7}{3}n$ vertices. This…

A strengthened version of Harborth's well-known conjecture -- known as Kleber's conjecture -- states that every planar graph admits a planar straight-line drawing where every edge has integer length and each vertex is restricted to the…

An edge-ordered graph is a graph with a total ordering of its edges. A path $P=v_1v_2\ldots v_k$ in an edge-ordered graph is called increasing if $(v_iv_{i+1}) > (v_{i+1}v_{i+2})$ for all $i = 1,\ldots,k-2$; it is called decreasing if…

Equitable list arboricity, introduced by Zhang in 2016, generalizes the notion of equitable list coloring by requiring the subgraph induced by each color class to be acyclic (instead of edgeless) in addition to the usual upper bound on the…

We start a systematic investigation concerning bipartite Tur\'an number for trees. For a graph $F$ and integers $1 \leq a \leq b$ we define: $(i)$\quad $ex_b(a, b, F)$ is the largest number of edges that an $F$-free bipartite graph can have…

In 1995, Meinardus & Berg presented a reformulation of the Collatz Conjecture in terms of a functional equation in a single complex variable over the open unit disk. This paper generalizes that method to deal with not only a large class of…

An operational approach to the Collatz Conjecture is presented. Scenarios are defined as strings of characters "s" (for "spike") and "d" (for "down") which symbolize the Collatz operations (3m+1)/2 and m/2 in a Collatz Series connecting two…

The infinite grid is the Cayley graph of $\mathbb{Z} \times \mathbb{Z}$ with the usual generators. In this paper, the Ihara zeta function for the infinite grid is computed using elliptic integrals and theta functions. The zeta function of…

This paper studies equable parallelograms whose vertices lie on the Eisenstein lattice. Using Rosenberger's Theorem on generalised Markov equations, we show that the set of these parallelograms forms naturally an infinite tree, all of whose…