Large cycles in 4-connected graphs

Every 4-connected graph $G$ with minimum degree $\delta$ and connectivity $\kappa$ either contains a cycle of length at least $4\delta-\kappa-4$ or every longest cycle in $G$ is a dominating cycle.Comment: 4 page

A Size Bound for Hamilton Cycles

Every graph of size $q$ (the number of edges) and minimum degree $\delta$ is hamiltonian if $q\le\delta^2+\delta-1$. The result is sharp.Comment: 3 page

A Note on Hamilton Cycles

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.Comment: 2 page

On kk-ended spanning and dominating trees

A tree with at most $k$ leaves is called a $k$-ended tree. A spanning 2-ended tree is a Hamilton path. A Hamilton cycle can be considered as a spanning 1-ended tree. The earliest result concerning spanning trees with few leaves states that if $k$ is a positive integer and $G$ is a connected graph of order $n$ with $d(x)+d(y)\ge n-k+1$ for each pair of nonadjacent vertices $x,y$, then $G$ has a spanning $k$-ended tree. In this paper, we improve this result in two ways, and an analogous result is proved for dominating $k$-ended trees based on the generalized parameter $t_k$ - the order of a largest $k$-ended tree. In particular, $t_1$ is the circumference (the length of a longest cycle), and $t_2$ is the order of a longest path.Comment: 7 page

A Common Generalization of Dirac's two Theorems

Let $G$ be a 2-connected graph of order $n$ and let $c$ be the circumference - the order of a longest cycle in $G$. In this paper we present a sharp lower bound for the circumference based on minimum degree $\delta$ and $p$ - the order of a longest path in $G$. This is a common generalization of two earlier classical results for 2-connected graphs due to Dirac: (i) $c\ge \min\{n,2\delta\}$; and (ii) $c\ge\sqrt{2p}$. Moreover, the result is stronger than (ii).Comment: 7 page

On Relative Length of Long Paths and Cycles in Graphs

Let $G$ be a graph on $n$ vertices, $p$ the order of a longest path and $\kappa$ the connectivity of $G$. In 1989, Bauer, Broersma Li and Veldman proved that if $G$ is a 2-connected graph with $d(x)+d(y)+d(z)\ge n+\kappa$ for all triples $x,y,z$ of independent vertices, then $G$ is hamiltonian. In this paper we improve this result by reducing the lower bound $n+\kappa$ to $p+\kappa$.Comment: 8 page

On Dirac's Conjecture

Let $G$ be a 2-connected graph, $l$ be the length of a longest path in $G$ and $c$ be the circumference - the length of a longest cycle in $G$. In 1952, Dirac proved that $c>\sqrt{2l}$ and conjectured that $c\ge 2\sqrt{l}$. In this paper we present more general sharp bounds in terms of $l$ and the length $m$ of a vine on a longest path in $G$ including Dirac's conjecture as a corollary: if $c=m+y+2$ (generally, $c\ge m+y+2$) for some integer $y\ge 0$, then $c\ge\sqrt{4l+(y+1)^2}$ if $m$ is odd; and $c\ge\sqrt{4l+(y+1)^2-1}$ if $m$ is even.Comment: 6 pages, major revisio

Disconnected Forbidden Subgraphs, Toughness and Hamilton Cycles

In 1974, Goodman and Hedetniemi proved that every 2-connected $(K_{1,3},K_{1,3}+e)$-free graph is hamiltonian. This result gave rise many other hamiltonicity conditions for various pairs and triples of forbidden connected subgraphs under additional connectivity conditions. In 1997, it was proved that a single forbidden connected subgraph $R$ in 2-connected graphs can create only a trivial class of hamiltonian graphs (complete graphs) with $R=P_3$. In this paper we prove that a single forbidden subgraph $R$ can create a non trivial class of hamiltonian graphs if $R$ is disconnected: $(\ast1)$ every $(K_1\cup P_2)$-free graph either is hamiltonian or belongs to a well defined class of non hamiltonian graphs; $(\ast2)$ every 1-tough $(K_1\cup P_3)$-free graph is hamiltonian. We conjecure that every 1-tough $(K_1\cup P_4)$-free graph is hamiltonian and every 1-tough $P_4$-free graph is hamiltonianComment: 6 pages, corrected and improve

Nonhamiltonian Graphs with Given Toughness

In 1973, Chv\'{a}tal introduced the concept of toughness $\tau$ of a graph and constructed an infinite class of nonhamiltonian graphs with $\tau=3/2$. Later Thomassen found nonhamiltonian graphs with $\tau>3/2$, and Enomoto et al. constructed nonhamiltonian graphs with $\tau=2-\epsilon$ for each positive $\epsilon$. The last result in this direction is due to Bauer, Broersma and Veldman, which states that for each positive $\epsilon$, there exists a nonhamiltonian graph with $\tau\ge 9/4-\epsilon$. In this paper we prove that for each rational number $t$ with $0<t<9/4$, there exists a nonhamiltonian graph with $\tau=t$.Comment: 11 pages, corrected and improved versio

On some Versions of Conjectures of Bondy and Jung

Using algebraic transformations and equivalent reformulations we derive a number of new results from some earlier ones (by the author) in more accepted terms closely related to well-known conjectures of Bondy and Jung including a number of classical results in hamiltonian graph theory (due to Dirac, Ore, Nash-Williams, Bondy, Jung and so on) as special cases. A number of extended and strengthened versions of these conjectures are proposed.Comment: 8 page