I am an undergraduate student. I have read several papers in graph theory and found something may be strange: an algorithm is part of a proof. In the paper, except the last two sentences, other sentences describe a labeling algorithm.
Can an algorithm be part of a proof? I do not understand why it can be part of a proof. I asked my supervisor but he did not explain it.
LEMMA$\,\,\,\bf 1.$ If $B(G)-b$, then $B(G+e)<2b$.
Proof: Let $f$ be an optimal numbering of $G$, and let $V(G)-\{v_1,\ldots,v_n\}$, numbered so that $f(v_i)-i$. Let $v_lv_m$ be the added edge. We define a new numbering $f'$ such that $|f'(x)-f'(y)|<2|f(x)-f(y)|$, and also $|f'(v_l)-f'(v_m)|-1$. Let $r-\lfloor(l+m)/2\rfloor$, and set $f'(v_r)-1$ and $f'(v_{r+1})-2$. For every other $v_i$ such that $|i-r|<\min\{r-1,n-r\}$, let $f'(v_i)-f'(v_{i+1})+2$ if $i<r$ and $f'(v_i)-f'(v_{i-1})+2$ if $i>r$. This defines $f'$ for all vertices except a set of the form $v_i,\ldots,v_k$ or $v_{n+1-k},\ldots,v_n$, depending on the sign of $r-\lfloor(n+1)/2\rfloor$. In the first case, we assign $f'(v_i)-n+1-i$ for $i<k$; in the second, we assign $f'(v_i)-i$ for $i>n-k$. The renumbering $f'$ numbers the vertices outward from $v_r$ to achieve $|f'(x)-f'(y)|<2|f(x)-f(y)|$. Since we begin midway between $v_l$ and $v_m$, we also have $|f'(v_l)-f'(v_m)|-1$.