Suppose that $T^{\circ }$ and $T^{☆}$ are partial latin squares of order $n$, with the property that each row and each column of $T^{\circ }$ contains the same set of entries as the corresponding row or column of $T^{☆}$. In addition, suppose that each cell in $T^{\circ }$ contains an entry if and only if the corresponding cell in $T^{☆}$ contains an entry, and these entries (if they exist) are different. Then the pair $T=(T^{\circ },T^{☆})$ forms a . The of $T$ is the total number of filled cells in $T^{\circ }$ (equivalently $T^{☆}$). The latin bitrade is if there is no...

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 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 <...

In [3], Faudree and Gould showed that if a 2-connected graph contains no $K_{1,3}$ and P₆ as an induced subgraph, then the graph is hamiltonian. In this paper, we consider the extension of this result to cycles passing through specified vertices. We define the families of graphs which are extension of the forbidden pair $K_{1,3}$ and P₆, and prove that the forbidden families implies the existence of cycles passing through specified vertices.

We prove that every vertex v of a tournament T belongs to at least $maxmin\delta ⁺\left(T\right),2\delta ⁺\left(T\right)-d{⁺}_T\left(v\right)+1,min\delta ¯\left(T\right),2\delta ¯\left(T\right)-d{¯}_T\left(v\right)+1$ arc-disjoint cycles, where δ⁺(T) (or δ¯(T)) is the minimum out-degree (resp. minimum in-degree) of T, and $d{⁺}_T\left(v\right)$ (or $d{¯}_T\left(v\right)$) is the out-degree (resp. in-degree) of v.

An orientation of a simple graph is referred to as an oriented graph. Caccetta and Häggkvist conjectured that any digraph on $n$ vertices with minimum outdegree $d$ contains a directed cycle of length at most $\lceil n/d\rceil$. In this paper, we consider short cycles in oriented graphs without directed triangles. Suppose that ${\alpha }_0$ is the smallest real such that every $n$-vertex digraph with minimum outdegree at least ${\alpha }_{0}n$ contains a directed triangle. Let $ϵ<(3-2{\alpha }_0)/(4-2{\alpha }_0)$ be a positive real. We show that if $D$ is an oriented graph...

We consider limit cycles of a class of polynomial differential systems of the form $$\left\{\begin{array}{c}\dot{x}=y,\hfill \\ \dot{y}=-x-\epsilon (g_{21}\left(x\right)y^{2\alpha +1}+f_{21}\left(x\right)y^{2\beta })-{\epsilon }^2(g_{22}\left(x\right)y^{2\alpha +1}+f_{22}\left(x\right)y^{2\beta }),\hfill \end{array}\right.$$ where $\beta$ and $\alpha$ are positive integers, $g_{2j}$ and $f_{2j}$ have degree $m$ and $n$, respectively, for each $j=1,2$, and $\epsilon$ is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center $\dot{x}=y$, $\dot{y}=-x$ using the averaging theory of first and second order.

Se il problema di minimo $({𝒫}_{\mathrm{\infty }})$ è il limite, in senso variazionale, di una successione di problemi di minimo con ostacoli del tipo $$\underset{u\ge {\psi }_h}{\min }{\int }_A\left[f_h(x,u,Du)+b(x,u)\right]𝑑x,$$ allora $({𝒫}_{\mathrm{\infty }})$ può essere scritto nella forma $$\underset{u}{\min }\left\{{\int }_A\left[f_{\mathrm{\infty }}(x,u,Du)+b(x,u)\right]𝑑x+{\int }_A{g}_{\mathrm{\infty }}(x,\tilde{u}(x))𝑑{\lambda }_{\mathrm{\infty }}(x)\right\}$$ dove $\tilde{u}$ è un conveniente rappresentante di $u$ e ${\lambda }_{\mathrm{\infty }}$ è una misura non negativa.

A proper vertex coloring of a graph $G$ is acyclic if there is no bicolored cycle in $G$. In other words, each cycle of $G$ must be colored with at least three colors. Given a list assignment $L=\left\{L\right(v):v\in V\}$, if there exists an acyclic coloring $\pi$ of $G$ such that $\pi \left(v\right)\in L\left(v\right)$ for all $v\in V$, then we say that $G$ is acyclically $L$-colorable. If $G$ is acyclically $L$-colorable for any list assignment $L$ with $\left|L\right(v\left)\right|\ge k$ for all $v\in V$, then $G$ is acyclically $k$-choosable. In 2006, Montassier, Raspaud and Wang conjectured that every planar graph without...

We consider the classical problem of a position of n-dimensional manifold Mⁿ in ${ℝ}^{n+2}$. We show that we can define the fundamental (n+1)-cycle and the shadow fundamental (n+2)-cycle for a fundamental quandle of a knotting $Mⁿ\to {ℝ}^{n+2}$. In particular, we show that for any fixed quandle, quandle coloring, and shadow quandle coloring, of a diagram of Mⁿ embedded in ${ℝ}^{n+2}$ we have (n+1)- and (n+2)-(co)cycle invariants (i.e. invariant under Roseman moves).