A Note on Repeated Degrees of Line Graphs

Let $\text{rep}(G)$ be the maximum multiplicity of a vertex degree in graph $G$. It was proven in Caro and West [E-JC, 2009] that if $G$ is an $n$-vertex line graph, then $\text{rep}(G) \geqslant \frac{1}{4} n^{1/3}$. In this note we prove that for infinitely many $n$ there is a $n$-vertex line graph $G$ such that $\text{rep}(G) \leqslant \left(2n\right)^{1/3}$, thus showing that the bound above is asymptotically tight. Previously it was only known that for infinitely many $n$ there is a $n$-vertex line graph $G$ such that $\text{rep}(G) \leqslant \sqrt{4n/3}$ (Caro and West [E-JC, 2009]). Finally we prove that if $G$ is a $n$-vertex line graph, then $\text{rep}(G) \geqslant \left(\left(\frac{1}{2}-o(1)\right)n\right)^{1/3}$.