Equating $k$ Maximum Degrees in Graphs without Short Cycles

Abstract

For an integer $k$ at least $2$, and a graph $G$, let $f_k(G)$ be the minimum cardinality of a set $X$ of vertices of $G$ such that $G-X$ has either $k$ vertices of maximum degree or order less than $k$. Caro and Yuster (Discrete Mathematics 310 (2010) 742-747) conjectured that, for every $k$, there is a constant $c_k$ such that $f_k(G)\leq c_k \sqrt{n(G)}$ for every graph $G$. Verifying a conjecture of Caro, Lauri, and Zarb (arXiv:1704.08472v1), we show the best possible result that, if $t$ is a positive integer, and $F$ is a forest of order at most $\frac{1}{6}\left(t^3+6t^2+17t+12\right)$, then $f_2(F)\leq t$. We study $f_3(F)$ for forests $F$ in more detail obtaining similar almost tight results, and we establish upper bounds on $f_k(G)$ for graphs $G$ of girth at least $5$. For graphs $G$ of girth more than $2p$, for $p$ at least $3$, our results imply $f_k(G)=O\left(n(G)^{\frac{p+1}{3p}}\right)$. Finally, we show that, for every fixed $k$, and every given forest $F$, the value of $f_k(F)$ can be determined in polynomial time.