Related papers: Circumference of 3-connected cubic graphs

The Erd\H{o}s--Gallai Theorem states that for $k\geq 3$ every graph on $n$ vertices with more than $\frac{1}{2}(k-1)(n-1)$ edges contains a cycle of length at least $k$. Kopylov proved a strengthening of this result for 2-connected graphs…

A non-planar graph is almost-planar if either deleting or contracting any edge makes it planar. A graph with $n$ vertices is pancyclic if it contains a cycle of every length from $3$ to $n$, and it is Hamiltonian if it contains a cycle of…

Harry Buhrman et al gave an Omega(sqrt n) lower bound for monotone graph properties in the adjacency matrix query model. Their proof is based on the polynomial method. However for some properties stronger lower bounds exist. We give an…

Li, Nikiforov and Schelp conjectured that a 2-edge coloured graph G with order n and minimal degree strictly greater than 3n/4 contains a monochromatic cycle of length l, for all l at least four and at most n/2. We prove this conjecture for…

An orbit of $G$ is a subset $S$ of $V(G)$ such that $\phi(u)=v$ for any two vertices $u,v\in S$, where $\phi$ is an isomorphism of $G$. The orbit number of a graph $G$, denoted by $\text{Orb}(G)$, is the number of orbits of $G$. In [A Note…

An edge subset \( S \subseteq E(G) \) is called a 3-restricted edge-cut if \( G - S \) is disconnected and each component of \( G - S \) contains at least three vertices. The 3-restricted edge-connectivity of a graph \( G \), denoted by \(…

Contraction of triangles is a standard operation in the study of cubic graphs, as it reduces the order of the graph while typically preserving many of its properties. In this paper, we investigate the converse problem, wherein certain…

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…

Bondy and Vince showed that every graph with minimum degree at least three contains two cycles of lengths differing by one or two.We prove the following average degree counterpart that every $n$-vertex graph $G$ with at least $\frac52(n-1)$…

A graph is called a $k$-planar unit distance graph if it can be drawn in the plane such that every edge is a unit line segment and is involved in at most $k$ crossings. We investigate $u_k(n)$, the maximum number of edges of such graphs on…

The unit distance graph $G_{\mathbb{R}^d}^1$ is the infinite graph whose nodes are points in $\mathbb{R}^d$, with an edge between two points if the Euclidean distance between these points is 1. The 2-dimensional version $G_{\mathbb{R}^2}^1$…

Twin-width is a recently introduced graph parameter based on the repeated contraction of near-twins. It has shown remarkable utility in algorithmic and structural graph theory, as well as in finite model theory -- particularly since…

There are many hard conjectures in graph theory, like Tutte's 5-flow conjecture, and the 5-cycle double cover conjecture, which would be true in general if they would be true for cubic graphs. Since most of them are trivially true for…

We show that for every cubic graph G with sufficiently large girth there exists a probability distribution on edge-cuts of G such that each edge is in a randomly chosen cut with probability at least 0.88672. This implies that G contains an…

We show that there exists an infinite family of cubic $2$-connected non-hamiltonian graphs with girth $5$ containing a unique longest cycle.

A cactus is a connected graph in which each edge is contained in at most one cycle. We generalize the concept of cactus graphs, i.e., a $k$-cactus is a connected graph in which each edge is contained in at most $k$ cycles where $k\ge 1$. It…

Dirac proved that each $n$-vertex $2$-connected graph with minimum degree at least $k$ contains a cycle of length at least $\min\{2k, n\}$. We consider a hypergraph version of this result. A Berge cycle in a hypergraph is an alternating…

A geometric graph is a graph drawn in the plane so that its vertices and edges are represented by points in general position and straight line segments, respectively. A vertex of a geometric graph is called pointed if it lies outside of the…

The cyclic edge-connectivity of a graph $G$ is the least $k$ such that there exists a set of $k$ edges whose removal disconnects $G$ into components where every component contains a cycle. We show that for graphs of minimum degree at least…

In this note we confirm a conjecture raised by Benjamini et al. \cite{BST} on the acquaintance time of graphs, proving that for all graphs $G$ with $n$ vertices it holds that $\AC(G) = O(n^{3/2})$, which is tight up to a multiplicative…