Efficient dominating sets in circulant graphs with domination number prime

Cayley graphs of finite cyclic group ${\mathbb{Z}}_n$ are called circulant graphs and denoted by $\mathrm{Cay}({\mathbb{Z}}_n,S)$. For $\mathrm{Cay}({\mathbb{Z}}_n,S)$ with $\frac{n}{|S|+1}$ prime, we give a necessary and sufficient condition for the existence of efficient dominating sets and characterize completely all its efficient dominating sets.     

Suslin’s algorithms for reduction of unimodular rows

A well-known lemma of Suslin says that for a commutative ring $A$ if $(v_1\left(X\right),\dots ,v_n\left(X\right))\in {\left(A\left[X\right]\right)}^n$ is unimodular where $v_1$ is monic and $n\ge 3$, then there exist ${\gamma }_1,\dots ,{\gamma }_{\ell }\in E_{n-1}\left(A\left[X\right]\right)$ such that the ideal generated by $\mathrm{Res}(v_1,e_1.{\gamma }_1{}^t(v_2,\dots ,v_n)),\dots ,\mathrm{Res}(v_1,e_1.{\gamma }_{\ell }{}^t(v_2,\dots ,v_n))$ equals $A$. This lemma played a central role in the resolution of Serre’s Conjecture. In the case where $A$ contains a set $E$ of cardinality greater than $deg{v}_1+1$ such that $y-y^{\prime }$ is invertible for each $y\ne y^{\prime }$ in $E$, we prove that the ${\gamma }_i$ can simply correspond to the elementary operations $L_1\to L_1+y_i{\sum }_{j=2}^{n-1}{u}_{j+1}{L}_j$, $1\le i\le \ell =deg{v}_1+1$, where $u_1{v}_1+\cdots +u_n{v}_n=1$. These efficient elementary operations enable us to give new and simple algorithms for reducing unimodular rows with entries in $K[X_1,\dots ,X_k]$ to ${}^t(1,0,\dots ,0)$ using elementary operations in the case where $K$ is an infinite field. Another feature of this paper is that it shows that the concrete local–global principles can produce competitive complexity bounds.     

Edge-fault-tolerant pancyclicity and bipancyclicity of Cartesian product graphs with faulty edges

Let r≥ 4 be an even integer. Graph G is r-bipancyclic if it contains a cycle of every even length from r to $2\lfloor \frac{n(G)}{2}\rfloor$, where $n(G)$ is the number of vertices in G. A graph G is r-pancyclic if it contains a cycle of every length from r to $n(G)$, where $r\ge 3$. A graph is k-edge-fault Hamiltonian if, after deleting arbitrary k edges from the graph, the resulting graph remains Hamiltonian. The terms k-edge-fault r-bipancyclic and k-edge-fault r-pancyclic can be defined similarly. Given two graphs G and H, where $n(G)$, $n(H)\ge$ 9, let $k_1$, $k_2\ge 5$ be the minimum degrees of G and H, respectively. This study determined the edge-fault r-bipancyclic and edge-fault r-pancyclic of Cartesian product graph $G\times H$ with some conditions. These results were then used to evaluate the edge-fault pancyclicity (bipancyclicity) of ${\mathit{NQ}}_{m_r,\dots ,m_1}$ and ${\mathit{GQ}}_{m_r,\dots ,m_1}$.