Related papers: A study of $4-$cycle systems

A $k$-uniform tight cycle is a $k$-uniform hypergraph with a cyclic ordering of its vertices such that its edges are all the sets of size $k$ formed by $k$ consecutive vertices in the ordering. We prove that every red-blue edge-coloured…

In this work we study the null space of bipartite graphs without cycles of length multiple of $4$, and its relation to structural properties. We decompose them into two subgraphs: $C_N(G)$ and $C_S(G)$. $C_N(G)$ has perfect matching and its…

A graph is said to be {\em half-arc-transitive} if its automorphism group acts transitively on the set of its vertices and edges but not on the set of its arcs. With each half-arc-transitive graph of valency 4 a collection of the so called…

Let $G$ be an edge-colored graph. A heterochromatic cycle of $G$ is one in which every two edges have different colors. For a vertex $v\in V(G)$, let $CN(v)$ denote the set of colors which are assigned to the edges incident to $v$. In this…

An avoidance problem of configurations in 4-cycle systems is investigated by generalizing the notion of sparseness, which is originally from Erd\H{o}s' r-sparse conjecture on Steiner triple systems. A 4-cycle system of order v, 4CS(v), is…

A diamond is the graph that is obtained from removing an edge from the complete graph on $4$ vertices. A ($C_4$,diamond)-free graph is a graph that does not contain a diamond or a cycle on four vertices as induced subgraphs. Let $G$ be a…

Let $G$ be a graph with edge set $(e_1,e_2,...e_N)$. We independently associate to each edge $e_i$ of $G$ a cost ${x}_i$ that is drawn from a Uniform [0, 1] distribution. Suppose $\mathcal{F}$ is a set of targeted structures that consists…

A separating system of a graph $G$ is a family $\mathcal{S}$ of subgraphs of $G$ for which the following holds: for all distinct edges $e$ and $f$ of $G$, there exists an element in $\mathcal{S}$ that contains $e$ but not $f$. Recently, it…

In a graph $G$, a subset of vertices $S \subseteq V(G)$ is said to be cyclable if there is a cycle containing the vertices in some order. $G$ is said to be $k$-cyclable if any subset of $k \geq 2$ vertices is cyclable. If any $k$…

Given an $r$-edge-colouring of the edges of a graph $G$, we say that it can be partitioned into $p$ monochromatic cycles when there exists a set of $p$ vertex-disjoint monochromatic cycles covering all the vertices of $G$. In the literature…

A 3-way $(v,k,t)$ trade $T$ of volume $m$ consists of three pairwise disjoint collections $T_1$, $T_2$ and $T_3$, each of $m$ blocks of size $k$, such that for every $t$-subset of $v$-set $V$, the number of blocks containing this $t$-subset…

A universal cycle is a compact listing of a class of combinatorial objects. In this paper, we prove the existence of universal cycles of classes of labeled graphs, including simple graphs, trees, graphs with m edges, graphs with loops,…

In this paper we investigate results of the form "every graph $G$ has a cycle $C$ such that the induced subgraph of $G$ on $V(G)\setminus V(C)$ has small maximum degree." Such results haven't been studied before, but are motivated by the…

We denote a path on $t$ vertices as $P_t$ and a cycle on $t$ vertices as $C_t$. For two vertex-disjoint graphs $G_1$ and $G_2$, the {\em union} $G_1\cup G_2$ is the graph with $V(G_1\cup G_2)=V(G_1)\cup V(G_2)$ and $E(G_1\cup…

It is shown that a hamiltonian $n/2$-regular bipartite graph $G$ of order $2n>8$ contains a cycle of length $2n-2$. Moreover, if such a cycle can be chosen to omit a pair of adjacent vertices, then $G$ is bipancyclic.

Let $f(x) \in \mathbb{Z}[x]$; for each integer $\alpha$ it is interesting to consider the number of iterates $n_{\alpha}$, if possible, needed to satisfy $f^{n_{\alpha}}(\alpha) = \alpha$. The sets $\{\alpha, f(\alpha), \ldots,…

For an $m$-cycle $C$, an inside $m$-cycle of $C$ is a cycle on the same vertex set, that is edge-disjoint from $C$. In an $m$-cycle system, $(\mathcal{X}, \mathcal{C})$, if inside $m$-cycles can be chosen -one for each cycle- to form…

The monography considers the problem of constructing a Hamiltonian cycle in a complete graph. A rule for constructing a Hamiltonian cycle based on isometric cycles of a graph is established. An algorithm for constructing a Hamiltonian cycle…

Cycle switching is a particular form of transformation applied to isomorphism classes of a Steiner triple system of a given order $v$ (an $STS(v)$), yielding another $STS(v)$. This relationship may be represented by an undirected graph. An…

An ${\ell}$-cycle system ${\mathcal F}$ of a graph $\Gamma$ is a set of ${\ell}$-cycles which partition the edge set of $\Gamma$. Two such cycle systems ${\mathcal F}$ and ${\mathcal F}'$ are said to be {\em orthogonal} if no two distinct…