Related papers: On cycles in intersection graph of rings

An edge of a graph of order $n$ is pancyclic if it lies in a cycle of every length $3,\ldots,n$. A graph of order $n$ is vertex-pancyclic if every vertex lies in a cycle of every length $3,\ldots,n$. Recently, Li and Zhan proved that every…

An $n$-vertex graph is Hamiltonian if it contains a cycle that covers all of its vertices, and it is pancyclic if it contains cycles of all lengths from $3$ up to $n$. In 1972, Erd\H{o}s conjectured that every Hamiltonian graph with…

We classify modules and rings with some specific properties of their intersection graphs. In particular, we describe rings with infinite intersection graphs containing maximal left ideals of finite degree. This answers a question raised in…

Let $G=(V,E)$ be a finite, simple graph. We consider for each oriented graph $G_{\cal O}$ associated to an orientation ${\cal O}$ of the edges of $G$, the toric ideal $P_{G_{\cal O}}$. In this paper we study those graphs with the property…

Let R be a commutative ring with identity. In this paper, we introduce and investigate the second ideal intersection graph SII(R) of R with vertices are non-zero proper ideals of R and two distinct vertices I and J are adjacent if and only…

We prove new lower bounds on the crossing number of a complete graphs assuming that it is drawn in such a way that it contains a Hamiltonian cycle with no crossings.

In this article, we introduce a new graph theoretic structure associated with a finite commutative ring, called nil clean divisor graph. For a ring $R$, nil clean divisor graph is denoted by $G_N(R)$, where the vertex set is $\{x\in R\,:\,…

A graph $G$ is called an $[s,t]$-graph if any induced subgraph of $G$ of order $s$ has size at least $t.$ We prove that every $2$-connected $[4,2]$-graph of order at least $7$ is pancyclic. This strengthens existing results. There are…

A conjecture of Carsten Thomassen states that every 4-connected line graph is hamiltonian. It is known that the conjecture is true for 7-connected line graphs. We improve this by showing that any 5-connected line graph of minimum degree at…

We consider the derangement graph in which the vertices are permutations of $\{ 1,\ldots, n\}$. Two vertices are joined by an edge if the corresponding permutations differ in every position. The derangement graph is known to be Hamiltonian…

We prove the following theorem. Let $r\ge 4$ be an integer, and $G$ be a $K_{1,r}$-free $r$-edge-connected $r$-regular graph. Then, for every set $W$ of even number of vertices of $G$ such that the distance between any two vertices of $W$…

In this paper, we characterize the positive integers $n$ for which intersection graph of ideals of $\mathbb{Z}_n$ is perfect.

We conjecture that the balanced complete bipartite graph $K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil}$ contains more cycles than any other $n$-vertex triangle-free graph, and we make some progress toward proving this. We give equivalent…

In 1966, Gallai asked whether all longest paths in a connected graph have a nonempty intersection. The answer to this question is not true in general and various counterexamples have been found. However, there is a positive solution to…

A graph G on n vertices is Hamiltonian if it contains a cycle of length n and pancyclic if it contains cycles of length $\ell$ for all $3 \le \ell \le n$. Write $\alpha(G)$ for the independence number of $G$, i.e. the size of the largest…

We consider complete intersection ideals in a polynomial ring over a field of characteristic zero that are stable under the action of the symmetric group permuting the variables. We determine the possible representation types for these…

For every graph $X$, we consider the class of all connected $\{K_{1,3}, X\}$-free graphs which are distinct from an odd cycle and have independence number at least $4$, and we show that all graphs in the class are perfect if and only if $X$…

A road interchange where $n$ roads meet and in which the drivers are not allowed to change lanes can be modelled as an embedding of a 2-coloured (hence bipartite) multigraph $G$ with equal-sized colour classes into an orientable surface…

Let $R$ be a commutative ring with $\Z(R)$ its set of zero-divisors. In this paper, we study the total graph of $R$, denoted by $\T(\Gamma(R))$. It is the (undirected) graph with all elements of $R$ as vertices, and for distinct $x, y\in…

Let $D$ be a division ring, $n$ a positive integer, and GL$_n(D)$ the general linear group of degree $n$ over $D$. In this paper, we study the induced subgraph of the intersection graph of GL$_n(D)$ generated by all non-trivial proper…