Some Generator Subgraphs of the Square of a Cycle

Abstract

Graphs considered in this paper are finite simple graphs, which has no loops and multiple edges. Let $G= (V(G),E(G))$ be a graph with $E(G) = \{e_{1}, e_{2},\ldots, e_{m}\},$  for some positive integer $m.$ The \textit{edge space} of $G,$ denoted by $\mathscr{E}(G),$  is a vector space over the  field $\zn_{2}.$  The elements of $\mathscr{E}(G)$ are all the  subsets of $E(G)$. Vector addition is defined as  $X+Y = X ~\Delta~ Y,$  the  symmetric difference of sets $X$ and $Y,$  for $X,Y \in \mathscr{E}(G).$ Scalar multiplication is defined as $1\cdot X =X$ and $0 \cdot X = \emptyset$ for  $X \in  \mathscr{E}(G).$ Let   $H$ be  a subgraph of $G.$ The \textit{uniform set of $H$} with respect to $G,$ denoted by $E_{H}(G),$ is the set of all elements of $\mathscr E(G)$ that induces a subgraph isomorphic to $H.$ The subspace of $\mathscr E(G)$ generated by  $E_{H}(G)$ shall be denoted by $\mathscr E_{H}(G).$ If $E_H(G)$ is a generating set, that is $\mathscr E_{H}(G)= \mathscr E(G),$ then $H$ is called a \textit{generator subgraph} of $G.$ This paper determines some generator subgraphs of the square of a cycle. Moreover, this paper established  sufficient  conditions for the generator subgraphs of the square of a cycle.