Every cyclic central division algebra \(A\) of prime degree \(p\) over a field \(F\) of characteristic \(p\) is generated by two elements \(x\), \(y\) subject to relations of the form \(x^p-x=a\), \(y^p=b\), and \(yxy^{-1}=x+1\), for some \(a,b\in F\), with \(b\neq 0\). Let \(X_A=\{x\in A\mid x^p-x\in F,\;x\notin F\}\) and \(Y_A=\{y\in A\mid y^p\in F,\;y\notin F\}\). The set \(X_A\cup Y_A\) may be viewed as the set of vertices of a bipartite graph whose edges are the pairs \(\{x,y\}\) such that \(yxy^{-1}=x+1\). The author investigates the properties of this graph. His main result is that if \(p=3\), then for any two elements \(x=x_0,z=x_3\in X_A\), there exist \(x_1,x_2\in X_A\) and \(y_1,y_2,y_3\in Y_A\) such that \(\{x_{i-1},y_i\}\) and \(\{x_i,y_i\}\) are vertices of the graph for \(i=1,2,3\). This result is inspired by an analogous ``chain-lemma'' for division algebras of degree~\(3\) over a field containing a primitive cube root of unity, due to \textit{M.~Rost} [C. R. Acad. Sci., Paris, Sér. I, Math. 328, No. 3, 185-190 (1999; Zbl 0934.12001)]. (Also submitted to MR).