Geodesic
Degree powers in graphs with a forbidden even cycle
The electronic journal of combinatorics, Tome 16 (2009) no. 1
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Let $C_{l}$ denote the cycle of length $l$. For $p\geq2$ and integer $k\geq1$, we prove that the function $$ \phi\left( k,p,n\right) =\max_G\left\{ \sum_{u\in V\left( G\right) } d^{p}\left( u\right)\right\} $$ (where the maximum is over graphs $G$ of order $n$ containing no $C_{2k+2}$) satisfies $\phi\left( k,p,n\right) =kn^{p}\left( 1+o\left( 1\right) \right)$. This settles a conjecture of Caro and Yuster. Our proof is based on a new sufficient condition for long paths.
DOI : 10.37236/196
Classification : 05C35, 05C38 
@article{10_37236_196,
     author = {Vladimir Nikiforov},
     title = {Degree powers in graphs with a forbidden even cycle},
     journal = {The electronic journal of combinatorics},
     year = {2009},
     volume = {16},
     number = {1},
     doi = {10.37236/196},
     zbl = {1186.05074},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/196/}
}
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Vladimir Nikiforov. Degree powers in graphs with a forbidden even cycle. The electronic journal of combinatorics, Tome 16 (2009) no. 1. doi: 10.37236/196

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