  Let \(G\) be a finite solvable group, and let \(\Delta(G)\) denote the
\emph{prime graph} built on the set of degrees of the irreducible complex
characters of \(G\). A fundamental result by P.P. P\'alfy asserts that the
complement $\bar{\Delta}(G)$ of the graph \(\Delta(G)\) does not contain any
cycle of length \(3\). In this paper we generalize P\'alfy's result, showing
that $\bar{\Delta}(G)$ does not contain any cycle of odd length, whence it is a
bipartite graph. As an immediate consequence, the set of vertices of
\(\Delta(G)\) can be covered by two subsets, each inducing a complete subgraph.
The latter property yields in turn that if \(n\) is the clique number of
\(\Delta(G)\), then \(\Delta(G)\) has at most \(2n\) vertices. This confirms a
conjecture by Z. Akhlaghi and H.P. Tong-Viet, and provides some evidence for
the famous \emph{\(\rho\)-\(\sigma\) conjecture} by B. Huppert.
