A short note on $K_{n}$-irregular graphs

Source abstract

This note addresses Theorem 12 in the seminal paper ``$F$-degrees in graphs'' by G. Chartrand et al. (1987), which asserts the existence of $K_n$-irregular graphs for each $n \ge 3$. Although this result is widely cited in the literature as an established fact, the original text lacks a complete proof. We show that for $n = 5$ and all $n \ge 7$, the constructions as presented are not $K_n$-irregular, presumably due to typographical errors. Consequently, referencing this foundational work as containing a rigorous proof for the general case is not entirely accurate.