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arXiv · 2609.30355

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

Tatiana Dovzhenok
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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.

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Tatiana Dovzhenok. 2026-09-24. A short note on $K_{n}$-irregular graphs. https://arxiv.org/abs/2609.30355

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