The Euclidean dimension of the join of two cycles (Q998454)

Suppose \(G\) is a graph that can be embedded in Euclidean space {\textbf{R}}\(^p\) in such a way that two vertices of \(G\) are adjacent if and only if they are unit distance apart in the embedding in {\textbf{R}}\(^p\). If such an embedding exists, then \(G\) is said to have unit-distance representation in {\textbf{R}}\(^p\). The smallest \(p\) such that \(G\) has unit-distance representation in {\textbf{R}}\(^p\) is called the dimension of \(G\), denoted by \(dim(G)\). The join of two graphs \(G\) and \(H\), denoted by \(G+H\) is the graph obtained from \(G\) and \(H\) by joining every vertex of \(G\) with every vertex of \(H\). It is shown that \(\dim(C_4+C_4)=4\), \(\dim(C_5+C_5)=4\), \(\dim(C_6+C_6)\) and for all all other values of \(m,n \geq 3\), \(\dim(C_m+C_n) = 5\).