Related papers: Cuts in Cartesian Products of Graphs

Consider a projective limit G of finite groups G_n. Fix a compatible family \delta^n of coactions of the G_n on a C*-algebra A. From this data we obtain a coaction \delta of G on A. We show that the coaction crossed product of A by \delta…

A $k$-box $B=(R_1,...,R_k)$, where each $R_i$ is a closed interval on the real line, is defined to be the Cartesian product $R_1\times R_2\times ...\times R_k$. If each $R_i$ is a unit length interval, we call $B$ a $k$-cube. Boxicity of a…

The Haagerup property, which is a strong converse of Kazhdan's property $(T)$, has translations and applications in various fields of mathematics such as representation theory, harmonic analysis, operator K-theory and so on. Moreover, this…

We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex $Q^-_3$ (here contraction means contracting the edges…

This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least $2k$ vertices is {\it $k$-linked} if, for every set of $2k$ distinct vertices organised in arbitrary $k$ pairs of vertices, there…

The strong geodetic problem is a recent variation of the geodetic problem. For a graph $G$, its strong geodetic number ${\rm sg}(G)$ is the cardinality of a smallest vertex subset $S$, such that each vertex of $G$ lies on a fixed shortest…

Determining the minimum genus of a graph is a fundamental optimisation problem in the study of network design and implementation as it gives a measure of non-planarity of graphs. In this paper, we are concerned with determining the smallest…

Recently, Bapat and Kurata [\textit{Linear Algebra Appl.}, 562(2019), 135-153] defined the Cartesian product of two square matrices $A$ and $B$ as $A\oslash B=A\otimes \J+\J\otimes B$, where $\J$ is the all one matrix of appropriate order…

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. An edge subset $F\subseteq E(G)$ is called a restricted edge-cut if $G-F$ is disconnected and has no isolated vertices. The restricted edge-connectivity $\lambda'(G)$ of $G$ is…

The generalized $k$-connectivity $\kappa_{k}(G)$ of a graph $G$, which was introduced by Chartrand et al.(1984) is a generalization of the concept of vertex connectivity. Let $G$ and $H$ be nontrivial connected graphs. Recently, Li et al.…

We present a simple, natural #P-complete problem. Let G be a directed graph, and let k be a positive integer. We define q(G;k) as follows. At each vertex v, we place a k-dimensional complex vector x_v. We take the product, over all edges…

k-graphs are higher-rank analogues of directed graphs which were first developed to provide combinatorial models for operator algebras of Cuntz-Krieger type. Here we develop the theory of covering spaces for k-graphs, obtaining a…

Suppose $k$ is a finite field, that $C$ is a smooth projective geometrically irreducible curve over $k$, and that $n$ is a positive integer not divisible by the characteristic of $k$. In this paper we compute cup products of elements of the…

The thickness $\theta(G)$ of a graph $G$ is the minimum number of planar spanning subgraphs into which the graph $G$ can be decomposed. It is a topological invariant of a graph, which was defined by W.T. Tutte in 1963 and also has important…

We consider the problem of extending partial edge colorings of cartesian products of graphs. More specifically, we suggest the following Evans-type conjecture: If $G$ is a graph where every precoloring of at most $k$ precolored edges can be…

This paper shows that $K_t$-minor-free (and $K_{s, t}$-minor-free) graphs $G$ are subgraphs of products of a tree-like graph $H$ (of bounded treewidth) and a complete graph $K_m$. Our results include optimal bounds on the treewidth of $H$…

Let $k$ be a positive integer and let $G$ be a graph with $n$ vertices. A connected $k$-subpartition of $G$ is a collection of $k$ pairwise disjoint sets (a.k.a. classes) of vertices in $G$ such that each set induces a connected subgraph.…

Graph product structure theory expresses certain graphs as subgraphs of the strong product of much simpler graphs. In particular, an elegant formulation for the corresponding structural theorems involves the strong product of a path and of…

Let $K_n^h=(V,\binom{V}{h})$ be the complete $h$-uniform hypergraph on vertex set $V$ with $|V|=n$. Baranyai showed that $K_n^h$ can be expressed as the union of edge-disjoint $r$-regular factors if and only if $h$ divides $rn$ and $r$…

We study the free product of rooted graphs and its various decompositions using quantum probabilistic methods. We show that the free product of rooted graphs is canonically associated with free independence, which completes the proof of the…