Related papers: $L(n)$ graphs are vertex-pancyclic and Hamilton-co…

A supergrid graph is a finite vertex-induced subgraph of the infinite graph whose vertex set consists of all points of the plane with integer coordinates and in which two vertices are adjacent if the difference of their x or y coordinates…

A graph is \emph{hamiltonian-connected} if every pair of vertices can be connected by a hamiltonian path, and it is \emph{hamiltonian} if it contains a hamiltonian cycle. We construct families of non-hamiltonian graphs for which the ratio…

Kronk introduced the $l$-path hamiltonianicity of graphs in 1969. A graph is $l$-path Hamiltonian if every path of length not exceeding $l$ is contained in a Hamiltonian cycle. We have shown that if $P=uvz$ is a 2-path of a 2-connected,…

The generalized Petersen graph $G(n, k)$ is a cubic graph with vertex set $V(G(n, k)) = \{v_i\}_{0 \leq i < n} \cup \{w_i\}_{0 \leq i < n}$ and edge set $E(G(n, k)) = \{v_i v_{i+1}\}_{0 \leq i < n} \cup \{w_i w_{i+k}\}_{0 \leq i < n} \cup…

A graph $G$ is Hamiltonian-connected if there exists a Hamiltonian path between any two vertices of $G$. It is known that if $G$ is 2-connected then the graph $G^2$ is Hamiltonian-connected. In this paper we prove that the square of every…

An $n$-vertex graph is called pancyclic if it contains a cycle of length $t$ for all $3 \leq t \leq n$. In this paper, we study pancyclicity of random graphs in the context of resilience, and prove that if $p \gg n^{-1/2}$, then the random…

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…

Let $G$ be a graph on $n\geq 3$ vertices, claw the bipartite graph $K_{1,3}$, and $Z_i$ the graph obtained from a triangle by attaching a path of length $i$ to its one vertex. $G$ is called 1-heavy if at least one end vertex of each induced…

A graph with $v$ vertices is $(r)$-pancyclic if it contains precisely $r$ cycles of every length from 3 to $v$. A bipartite graph with even number of vertices $v$ is said to be $(r)$-bipancyclic if it contains precisely $r$ cycles of each…

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 say that a k-uniform hypergraph C is an l-cycle if there exists a cyclic ordering of the vertices of C such that every edge of C consists of k consecutive vertices and such that every pair of consecutive edges (in the natural ordering of…

An oriented graph is an orientation of a simple graph. In 2009, Keevash, K\"{u}hn and Osthus proved that every sufficiently large oriented graph $D$ of order $n$ with $(3n-4)/8$ is Hamiltonian. Later, Kelly, K\"{u}hn and Osthus showed that…

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple $(v,u,x,y)$ of vertices such that both $(v,u,x)$ and $(u,x,y)$ are paths of length two. The 3-arc graph of a graph $G$ is defined to have vertices the arcs of $G$ such that two…

Let $H$ and $G$ be graphs such that $H$ has at least 3 vertices and is connected. The $H$-line graph of $G$, denoted by $HL(G)$, is that graph whose vertices are the edges of $G$ and where two vertices of $HL(G)$ are adjacent if they are…

For a graph $G$ with vertex set $V$, let N($G$) denote the number of nonempty subsets of $V$ that induce a connected graph in $G$. In this paper, we focus on determining N($G$) for $G$ in the family $\mathbb{B}_n$ of $n$-vertex bicyclic…

A graph $G$ of order $n$ is called edge-pancyclic if, for every integer $k$ with $3 \leq k \leq n$, every edge of $G$ lies in a cycle of length $k$. Determining the minimum size $f(n)$ of a simple edge-pancyclic graph with $n$ vertices…

A graph $G$ $=$ $(V,E)$ is vertex-pancyclic if for every vertex $u$ and any integer $l$ ranging from $3$ to $|V|$, $G$ contains a cycle $C$ of length $l$ such that $u$ is on $C$. A bipartite graph $G$ $=$ $(V,E)$ is vertex-bipancyclic if…

A graph $G$ is called an $[s,t]$-graph if any induced subgraph of $G$ of order $s$ has size at least $t$. In 2024, Zhan conjectured that every $2$-connected $[p + 2, p]$-graph of order at least $2p + 3$ and with minimum degree at least $p$…

A digraph $D=(V,A)$ of order $n\geq 3$ is pancyclic, whenever $D$ contains a directed cycle of length $k$ for each $k\in\{3,...,n\}$; and D is vertex-pancyclic iff, for each vertex $v\in V$ and each $k\in \{3,...,n\}$, $D$ contains a…

If $G$ is a more than one tough graph on $n$ vertices with $\delta\ge \frac{n}{2}-a$ for a given $a>0$ and $n$ is large enough then $G$ is hamiltonian.