Let T be a hamiltonian bipartite tournament with n vertices, γ a hamiltonian directed cycle of T, and k an even number. In this paper, the following question is studied: What is the maximum intersection with γ of a directed cycle of length k? It is proved that for an even k in the range 4 ≤ k ≤ [(n+4)/2], there exists a directed cycle $C_{h\left(k\right)}$ of length h(k), h(k) ∈ k,k-2 with $|A\left(C_{h\left(k\right)}\right)\cap A\left(\gamma \right)|\ge h\left(k\right)-3$ and the result is best possible. In a forthcoming paper the case of directed cycles of length k, k even and k <...

Let T be a hamiltonian bipartite tournament with n vertices, γ a hamiltonian directed cycle of T, and k an even number. In this paper the following question is studied: What is the maximum intersection with γ of a directed cycle of length k contained in T[V(γ)]? It is proved that for an even k in the range (n+6)/2 ≤ k ≤ n-2, there exists a directed cycle $C_{h\left(k\right)}$ of length h(k), h(k) ∈ k,k-2 with $|A\left(C_{h\left(k\right)}\right)\cap A\left(\gamma \right)|\ge h\left(k\right)-4$ and the result is best possible. In a previous paper a similar result for 4 ≤ k ≤ (n+4)/2 was...

Let T be a hamiltonian tournament with n vertices and γ a hamiltonian cycle of T. In previous works we introduced and studied the concept of cycle-pancyclism to capture the following question: What is the maximum intersection with γ of a cycle of length k? More precisely, for a cycle Cₖ of length k in T we denote $I_{\gamma }\left(Cₖ\right)=\left|A\left(\gamma \right)\cap A\left(Cₖ\right)\right|$, the number of arcs that γ and Cₖ have in common. Let $f(k,T,\gamma )=max{I}_{\gamma }\left(Cₖ\right)|Cₖ\subset T$ and f(n,k) = minf(k,T,γ)|T is a hamiltonian tournament with n vertices, and γ a hamiltonian cycle of T. In previous...

The basis number of a graph $G$ is defined by Schmeichel to be the least integer $h$ such that $G$ has an $h$-fold basis for its cycle space. MacLane showed that a graph is planar if and only if its basis number is $\le 2$. Schmeichel proved that the basis number of the complete graph $K_n$ is at most $3$. We generalize the result of Schmeichel by showing that the basis number of the $d$-th power of $K_n$ is at most $2d+1$.

In the paper, we give an existence theorem of periodic solution for Liénard equation $\dot{x}=y-F\left(x\right)$, $\dot{y}=-g\left(x\right)$. As a result, we estimate the amplitude $\rho \left(\mu \right)$ (maximal $x$-value) of the limit cycle of the van der Pol equation $\dot{x}=y-\mu (x^3/3-x)$, $\dot{y}=-x$ from above by $\rho \left(\mu \right)<2.3439$ for every $\mu \ne 0$. The result is an improvement of the author’s previous estimation $\rho \left(\mu \right)<2.5425$.

Let ${\mathbb{Z}}_n\left[\mathrm{i}\right]$ be the ring of Gaussian integers modulo $n$. We construct for ${\mathbb{Z}}_n\left[\mathrm{i}\right]$ a cubic mapping graph $\Gamma \left(n\right)$ whose vertex set is all the elements of ${\mathbb{Z}}_n\left[\mathrm{i}\right]$ and for which there is a directed edge from $a\in {\mathbb{Z}}_n\left[\mathrm{i}\right]$ to $b\in {\mathbb{Z}}_n\left[\mathrm{i}\right]$ if $b=a^3$. This article investigates in detail the structure of $\Gamma \left(n\right)$. We give suffcient and necessary conditions for the existence of cycles with length $t$. The number of $t$-cycles in ${\Gamma }_1\left(n\right)$ is obtained and we also examine when a vertex lies on a $t$-cycle of ${\Gamma }_2\left(n\right)$, where ${\Gamma }_1\left(n\right)$ is induced by all the units of ${\mathbb{Z}}_n\left[\mathrm{i}\right]$ while ${\Gamma }_2\left(n\right)$ is induced...