### Skolem-type difference sets for cycle systems.

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A graph H is obtained from a graph G by an edge rotation if G contains three distinct vertices u,v, and w such that uv ∈ E(G), uw ∉ E(G), and H = G-uv+uw. A graph H is obtained from a graph G by an edge jump if G contains four distinct vertices u,v,w, and x such that uv ∈ E(G), wx∉ E(G), and H = G-uv+wx. If a graph H is obtained from a graph G by a sequence of edge jumps, then G is said to be j-transformed into H. It is shown that for every two graphs G and H of the same order (at least 5) and same...

The (directed) distance from a vertex $u$ to a vertex $v$ in a strong digraph $D$ is the length of a shortest $u$-$v$ (directed) path in $D$. The eccentricity of a vertex $v$ of $D$ is the distance from $v$ to a vertex furthest from $v$ in $D$. The radius rad$D$ is the minimum eccentricity among the vertices of $D$ and the diameter diam$D$ is the maximum eccentricity. A central vertex is a vertex with eccentricity $\mathrm{r}adD$ and the subdigraph induced by the central vertices is the center $C\left(D\right)$. For a central vertex $v$ in a strong digraph...

A graph $G$ is a stratified graph if its vertex set is partitioned into classes (each of which is a stratum or a color class). A stratified graph with $k$ strata is $k$-stratified. If $G$ is a connected $k$-stratified graph with strata ${S}_{i}$ $(1\le i\le k)$ where the vertices of ${S}_{i}$ are colored ${X}_{i}$ $(1\le i\le k)$, then the ${X}_{i}$-proximity ${\rho}_{{X}_{i}}\left(v\right)$ of a vertex $v$ of $G$ is the distance between $v$ and a vertex of ${S}_{i}$ closest to $v$. The strati-eccentricity $se\left(v\right)$ of $v$ is $max\{{\rho}_{{X}_{i}}\left(v\right)\mid 1\le i\le k\}$. The minimum strati-eccentricity over all vertices of $G$ is the...

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