This question is about section 2.3 of Wilf's ``Algorithms and Complexity''
https://www.math.upenn.edu/~wilf/AlgoComp.pdf
in which he analyses the complexity of a recursive computation of the chromatic polynomial of a graph, based on the relation \begin{equation} \label{eq: recurrence} P(K;G)=P(K;G-\{e\})-P(K;G/\{e\}) \qquad (1). \end{equation} Here $P(K;G)$ is the number of ``proper colourings'' of $G$ with (as I understand it) at most $K$ colours (that is, colourings of the vertices using at most $K$ colours, such that no adjacent pair of vertices shares the same colour). The graph $G-\{e\}$ is the result of deleting edge $e$ from graph $G$, and the graph $G/\{e\}$ is the result of contracting edge $e$ in graph $G$. It turns out that $P(K;G)$ is indeed a polynomial in $K$, of degree equal to the number of vertices of $G$.
By defining $F(V,E)$ as the maximum cost (over all such graphs) of applying to a graph $G$ with at most $V$ vertices and at most $E$ edges, this recursive algorithm to compute $P(K;G)$, Wilf deduces from (1) that \begin{equation} \label{eq: F recurrence} F(V,E) \leq F(V,E-1)+cE+F(V-1,E-1) \qquad (2), \end{equation} with $F(V,0)=0$. Here the term $cE$ accounts for the effort of contracting the edge $e$, which requires us to modify the descriptions of any edges adjacent to the vertex that gets deleted.
Wilf seeks an upper bound of the form $F(V,E)\leq f(E)c$, and notes that the unique $f$ satisfying $f(E)=2f(E-1)+E$ and $f(0)=0$ will provide such a bound. He computes the exact solution $f$ of this last recurrence relation, which solution he approximates by $f(E)\sim2^{E+1}$.
He then derives another upper bound on $F$, by defining $\gamma(G)=|V(G)|+|E(G)|$, the sum of the number of vertices and the number of edges of $G$, and observing that $\gamma(G-\{e\})=\gamma(G)-1$ and $\gamma(G/\{e\})\leq \gamma(G)-2$. Then, letting $h(\gamma)$ denote the maximum amount of work done by the algorithm on any graph $G$ for which $\gamma(G)\leq \gamma$, Wilf claims that \begin{equation} \label{eq: h recurrence} h(\gamma)\leq h(\gamma-1)+h(\gamma-2) \qquad (3) \end{equation} for $\gamma \geq 2$. With $h(0)=h(1)=1$, the solution of this last recurrent inequality is that $h(\gamma) \leq F_{\gamma}$, the Fibonacci number.
Finally, combining the two bounds, he obtains that the time complexity of the algorithm is \begin{equation} O \left ( \min(2^{|E(G)|}, \phi^{|V(G)|+|E(G)|}) \right ), \end{equation} where $\phi$ is the golden ratio. Provided the ratio of $|E(G)|$ to $|V(G)|$ is large enough, the second bound is sharper than the first.
My question is, how in (3) was Wilf able to omit a term like $cE$ from (2) ?