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 $E(G)=\{e_1,e_2,\dots ,e_m\},$  for some positive integer $m.$ The \textit{edge space} of $G,$ denoted by $\mathcal{E}(G),$  is a vector space over the  field ${\text{zn}}_2.$  The elements of $\mathcal{E}(G)$ are all the  subsets of $E(G)$. Vector addition is defined as  $X+Y=X\mathrm{\Delta }Y,$  the  symmetric difference of sets $X$ and $Y,$  for $X,Y\in \mathcal{E}(G).$ Scalar multiplication is defined as $1\cdot X=X$ and $0\cdot X=\mathrm{\varnothing }$ for  $X\in \mathcal{E}(G).$ 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 $\mathcal{E}(G)$ that induces a subgraph isomorphic to $H.$ The subspace of $\mathcal{E}(G)$ generated by  $E_H(G)$ shall be denoted by ${\mathcal{E}}_H(G).$ If $E_H(G)$ is a generating set, that is ${\mathcal{E}}_H(G)=\mathcal{E}(G),$ 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.