Related papers: Circuit Preserving Edge Maps II

Let $M$ be a $3$-connected matroid. A pair $\{e,f\}$ in $M$ is detachable if $M \backslash e \backslash f$ or $M / e / f$ is $3$-connected. Williams (2015) proved that if $M$ has at least 13 elements, then at least one of the following…

Conduction graphs are defined here in order to elucidate at a glance the often complicated conduction behaviour of molecular graphs as ballistic molecular conductors. The graph $G^{\mathrm C}$ describes all possible conducting devices…

The Topological Representation Theorem for (oriented) matroids states that every (oriented) matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a homotopy sphere. In this paper, we…

If a graph $G_M$ is embedded into a closed surface $S$ such that $S \backslash G_M$ is a collection of disjoint open discs, then $M=(G_M,S)$ is called a {\em map}. A {\em zigzag} in a map $M$ is a closed path which alternates choosing, at…

For each odd integer $k\ge 5$, we prove that, if $M$ is a simple rank-$r$ binary matroid with no odd circuit of length less than $k$ and with $|M| > k 2^{r-k+1}$, then $M$ is isomorphic to a restriction of the rank-$r$ binary affine…

A {\em hole} is a chordless cycle of length at least four. A hole is {\em even} (resp. {\em odd}) if it contains an even (resp. odd) number of vertices. A \emph{cap} is a graph induced by a hole with an additional vertex that is adjacent to…

This paper presents sufficient conditions for Hamiltonian paths and cycles in graphs. Letting $\lambda\left( G\right) $ denote the spectral radius of the adjacency matrix of a graph $G,$ the main results of the paper are: (1) Let $k\geq1,$…

A graph is Hamiltonian if it contains a cycle passing through every vertex exactly once. A celebrated theorem of Dirac from 1952 asserts that every graph on $n\ge 3$ vertices with minimum degree at least $n/2$ is Hamiltonian. We refer to…

A graph $\textit{G}$ is a tuple $(\textit{V}, \textit{E})$, where $\textit{V}$ is the vertex set, $\textit{E}$ is the edge set. A reduced graph is a graph of deleting non-Hamiltonian edges and smoothing out the redundant vertices of degree…

In this paper we present some results for a connected infinite graph $G$ with finite degrees where the properties of balls of small radii guarantee the existence of some Hamiltonian and connectivity properties of $G$. (For a vertex $w$ of a…

A graph $G$ is $k$-edge-Hamiltonian if any collection of vertex-disjoint paths with at most $k$ edges altogether belong to a Hamiltonian cycle in $G$. A graph $G$ is $k$-Hamiltonian if for all $S\subseteq V(G)$ with $|S|\le k$, the subgraph…

We prove that if a unimodular random graph is almost surely planar and has finite expected degree, then it has a combinatorial embedding into the plane which is also unimodular. This implies the claim in the title immediately by a theorem…

The renowned theorem of Dirac states that if $G$ is a graph with minimum degree at least $n/2$ then $G$ has a Hamilton cycle. A natural generalisation asks what properties of an edge-colouring of $G$ guarantee the existence of a properly…

We look at the question of which distance-regular graphs are core-complete, meaning they are isomorphic to their own core or have a complete core. We build on Roberson's homomorphism matrix approach by which method he proved the…

A connected subgraph of a graph is isometric if it preserves distances. In this short note, we provide counterexamples to several variants of the following general question: When a graph $G$ is edge covered by connected isometric subgraphs…

A graph is one-ended if it contains a ray (a one way infinite path) and whenever we remove a finite number of vertices from the graph then what remains has only one component which contains rays. A vertex $v$ {\em dominates} a ray in the…

It is proven that for any integer $g \ge 0$ and $k \in \{ 0, \ldots, 10 \}$, there exist infinitely many 5-regular graphs of genus $g$ containing a 1-factorisation with exactly $k$ pairs of 1-factors that are perfect, i.e. form a…

A Hamiltonian embedding is an embedding of a graph $G$ such that the boundary of each face is a Hamiltonian cycle of $G$. It is shown that the hypercube graph $Q_n$ admits such an embedding on an orientable surface when $n$ is a power of 2.…

Let $G$ be a bridgeless cubic graph. The Berge--Fulkerson Conjecture (1970s) states that $G$ admits a list of six perfect matchings such that each edge of $G$ belongs to exactly two of these perfect matchings. If answered in the…

Given an infinite connected graph $G$, a way to randomly perturb its metric is to assign random i.i.d. lengths to the edges of the graph, a process called first-passage percolation. Assume that the graph is infinite and of bounded degree.…