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, Erdős conjectured that every Hamiltonian graph with independence number at most $k$ and at least $n = Ω(k^2)$ vertices is pancyclic. In this paper we prove this old conjecture in a strong form by showing that if such a graph has $n = (2+o(1))k^2$ vertices, it is already pancyclic, and this bound is asymptotically best possible.
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 $n$. A celebrated meta-conjecture of Bondy states that every non-trivial condition implying Hamiltonicity also implies pancyclicity (up to possibly a few exceptional graphs). We show that every graph $G$ with $κ(G) > (1+o(1)) α(G)$ is pancyclic. This extends the famous Chvátal-Erdős condition for Hamiltonicity and proves asymptotically a $30$-year old conjecture of Jackson and Ordaz.
The bipartite independence number of a graph $G$, denoted as $\tildeα(G)$, is the minimal number $k$ such that there exist positive integers $a$ and $b$ with $a+b=k+1$ with the property that for any two sets $A,B\subseteq V(G)$ with $|A|=a$ and $|B|=b$, there is an edge between $A$ and $B$. McDiarmid and Yolov showed that if $δ(G)\geq\tilde α(G)$ then $G$ is Hamiltonian, extending the famous theorem of Dirac which states that if $δ(G)\geq |G|/2$ then $G$ is Hamiltonian. In 1973, Bondy showed that, unless $G$ is a complete bipartite graph, Dirac's Hamiltonicity condition also implies pancyclici…
Let $G_1,...,G_n$ be graphs on the same vertex set of size $n$, each graph with minimum degree $δ(G_i)\ge n/2$. A recent conjecture of Aharoni asserts that there exists a rainbow Hamiltonian cycle i.e. a cycle with edge set $\{e_1,...,e_n\}$ such that $e_i\in E(G_i)$ for $1\leq i \leq n$. This can be viewed as a rainbow version of the well-known Dirac theorem. In this paper, we prove this conjecture asymptotically by showing that for every $\varepsilon>0$, there exists an integer $N>0$, such that when $n>N$ for any graphs $G_1,...,G_n$ on the same vertex set of size $n$ with $δ(G_i)\ge (\frac{…
In an edge-colored graph $(G,c)$, let $d^c(v)$ denote the number of colors on the edges incident with a vertex $v$ of $G$ and $δ^c(G)$ denote the minimum value of $d^c(v)$ over all vertices $v\in V(G)$. A cycle of $(G,c)$ is called proper if any two adjacent edges of the cycle have distinct colors. An edge-colored graph $(G,c)$ on $n\geq 3$ vertices is called properly vertex-pancyclic if each vertex of $(G,c)$ is contained in a proper cycle of length $\ell$ for every $\ell$ with $3 \le \ell \le n$. Fujita and Magnant conjectured that every edge-colored complete graph on $n\geq 3$ vertices with…
A graph $G$ on $m$ vertices is pancyclic if it contains cycles of length $l$, $3\leq l \leq m$ as subgraphs in $G$. The complete graph $K_{9}$ on 9 vertices with a cycle $C_{9}$ of length 9 deleted from $K_{9}$ is denoted by $(K_{9}-C_{9})$. In this paper, we prove that $(K_{9}-C_{9})^{n}$, the Cartesian product of $(K_{9}-C_{9})$ taken $n$ times, is pancyclic.
Let $G$ be an $n$-vertex graph, where $δ(G) \geq δn$ for some $δ:= δ(n)$. A result of Bohman, Frieze and Martin from 2003 asserts that if $α(G) = O \left(δ^2 n \right)$, then perturbing $G$ via the addition of $ω\left(\frac{\log(1/δ)}{δ^3} \right)$ random edges, asymptotically almost surely (a.a.s. hereafter) results in a Hamiltonian graph. This bound on the size of the random perturbation is only tight when $δ$ is independent of $n$ and deteriorates as to become uninformative when $δ= Ω\left(n^{-1/3} \right)$. We prove several improvements and extensions of the aforementioned result. First, k…
We study Hamiltonicity and pancyclicity in the graph obtained as the union of a deterministic $n$-vertex graph $H$ with $δ(H)\geqαn$ and a random $d$-regular graph $G$, for $d\in\{1,2\}$. When $G$ is a random $2$-regular graph, we prove that a.a.s. $H\cup G$ is pancyclic for all $α\in(0,1]$, and also extend our result to a range of sublinear degrees. When $G$ is a random $1$-regular graph, we prove that a.a.s. $H\cup G$ is pancyclic for all $α\in(\sqrt{2}-1,1]$, and this result is best possible. Furthermore, we show that this bound on $δ(H)$ is only needed when $H$ is `far' from containing a p…
Xiong and Liu [L. Xiong and Z. Liu, Hamiltonian iterated line graphs, Discrete Math. 256 (2002) 407-422] gave a characterization of the graphs $G$ for which the $n$-th iterated line graph $L^n(G)$ is hamiltonian, for $n\ge2$. In this paper, we study the existence of a hamiltonian path in $L^n(G)$, and give a characterization of $G$ for which $L^n(G)$ has a hamiltonian path. As applications, we use this characterization to give several upper bounds on the hamiltonian path index of a graph.
We revisit results obtained in [F. Harary, U. Peled, Hamiltonian threshold graphs, Discrete Appl.~Math., 16 (1987), 11--15], where several necessary and necessary and sufficient conditions for a connected threshold graph to be Hamiltonian were obtained. We present these results in new forms, now stated in terms of structural parameters that uniquely define the threshold graph and we extend them to chain graphs. We also identify the chain graph with minimum number of Hamilton cycles within the class of Hamiltonian chain graphs of a given order.
The prism over a graph $G$ is the Cartesian product of $G$ with the complete graph on two vertices. A graph $G$ is prism-hamiltonian if the prism over $G$ is hamiltonian. We prove that every polyhedral graph (i.e. 3-connected planar graph) of minimum degree at least four is prism-hamiltonian.
