Author:
Bollobás Béla,Nikiforov Vladimir
Abstract
For every real $p>0$ and simple graph $G,$ set $$ f\left( p,G\right) =\sum_{u\in V\left( G\right) }d^{p}\left( u\right) , $$ and let $\phi\left( r,p,n\right) $ be the maximum of $f\left( p,G\right) $ taken over all $K_{r+1}$-free graphs $G$ of order $n.$ We prove that, if $0 < p < r,$ then$$ \phi\left( r,p,n\right) =f\left( p,T_{r}\left( n\right) \right) , $$ where $T_{r}\left( n\right) $ is the $r$-partite Turan graph of order $n$. For every $p\geq r+\left\lceil \sqrt{2r}\right\rceil $ and $n$ large, we show that$$ \phi\left( p,n,r\right) >\left( 1+\varepsilon\right) f\left( p,T_{r}\left( n\right) \right) $$ for some $\varepsilon=\varepsilon\left( r\right) >0.$ Our results settle two conjectures of Caro and Yuster.
Publisher
The Electronic Journal of Combinatorics
Subject
Computational Theory and Mathematics,Geometry and Topology,Theoretical Computer Science,Applied Mathematics,Discrete Mathematics and Combinatorics
Cited by
12 articles.
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