doi: 10.7763/IJAPM.2014.V4.285
On the Turán Numbers for Even Cycles
,
Yongqi Sun
,
- Beijing Jiaotong University, China
Abstract
For integers <span id="cke_bm_86S" style="display: none;"> </span><span id="cke_bm_84S" style="display: none;"> </span><em>k</em><span id="cke_bm_86E" style="display: none;"> </span><span id="cke_bm_84E" style="display: none;"> </span>≥2 and <span id="cke_bm_87S" style="display: none;"> </span><em>n</em><span id="cke_bm_87E" style="display: none;"> </span>≥2<em>k</em>+1, let <span id="cke_bm_88E" style="display: none;"> </span><span id="cke_bm_95S" style="display: none;"> </span><em>ex</em><span id="cke_bm_95E" style="display: none;"> </span> (<span id="cke_bm_85E" style="display: none;"> </span><span id="cke_bm_96S" style="display: none;"> </span><em>n</em><span id="cke_bm_96E" style="display: none;"> </span><span id="cke_bm_97S" style="display: none;"> </span>,<em> C</em><sub>2<span id="cke_bm_97E" style="display: none;"> </span><em>k</em></sub>) den<span id="cke_bm_98E" style="display: none;"> </span>ote the maximum number of edges in a <em>C</em><sub>2<em>k</em></sub>-free graph of order <em>n</em>, and EX (<em>n</em>,<em> C</em><sub>2<em>k</em></sub>) denote the set of all graphs with <em>ex</em> (<em>n</em>,<em> C</em><sub>2<em>k</em></sub>) edges. For <em>k</em>=3, it is well known that <em>ex</em> (<em>n</em>,<em> C</em><sub>2<em>k</em></sub>) > 0.5338<em>n</em><sup>1+1/<em>k</em></sup> for some large <em>n</em>. In this paper, we study the values of <em>ex</em> (<em>n</em>,<em> C</em><sub>2<em>k</em></sub>) for k≥3 when<em> n</em> is small, their lower bounds were given based the three graphs without <em>C</em><sub>2<em>k</em></sub>. The known result shows that it is the tight lower bound for <em>k</em>=3 and <em>n</em>=28, and we further conjecture that <em>ex</em> (<em>n</em>,<em> C</em><sub>2<em>k</em></sub>)=(2<span id="cke_bm_119S" style="display: none;"> </span><span id="cke_bm_116S" style="display: none;"> </span><em>k</em><span id="cke_bm_119E" style="display: none;"> </span><span id="cke_bm_116E" style="display: none;"> </span><span id="cke_bm_117S" style="display: none;"> </span>+1)(2<span id="cke_bm_120S" style="display: none;"> </span><em>k</em><span id="cke_bm_120E" style="display: none;"> </span>−1)(2<span id="cke_bm_121S" style="display: none;"> </span><span id="cke_bm_117E" style="display: none;"> </span><em>k</em><span id="cke_bm_121E" style="display: none;"> </span>−2)/2 for<span id="cke_bm_122S" style="display: none;"> </span> <em>n</em>=<span id="cke_bm_122E" style="display: none;"> </span>4<em>k</em>2−<span id="cke_bm_123E" style="display: none;"> </span><span id="cke_bm_118E" style="display: none;"> </span>2<em>k</em>−2 an<em>d k</em>≥4.
Keywords
- Extremal graph
- even cycle
- lower bounds
- Turán numbers
How to Cite
Rui Zhang, Yongqi Sun, and Yali Wu, "On the Turán Numbers for Even Cycles," International Journal of Applied Physics and Mathematics, vol. 4, no. 3, pp. 211-214, 2014. https://doi.org/ 10.7763/IJAPM.2014.V4.285
Copyright & License
Copyright © 2014 by the authors. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).