1 research outputs found
Equating Maximum Degrees in Graphs without Short Cycles
For an integer at least , and a graph , let be the minimum
cardinality of a set of vertices of such that has either
vertices of maximum degree or order less than . Caro and Yuster (Discrete
Mathematics 310 (2010) 742-747) conjectured that, for every , there is a
constant such that for every graph .
Verifying a conjecture of Caro, Lauri, and Zarb (arXiv:1704.08472v1), we show
the best possible result that, if is a positive integer, and is a
forest of order at most , then
. We study for forests in more detail obtaining
similar almost tight results, and we establish upper bounds on for
graphs of girth at least . For graphs of girth more than , for
at least , our results imply
. Finally, we show that, for every
fixed , and every given forest , the value of can be determined
in polynomial time