%0 Journal Article
%T The Planar Ramsey Numbers PR (K<sub>4</sub>-e, K<sub>l</sub>)
%A Yongqi Sun
%A Yali Wu
%A Rui Zhang
%A Yuansheng Yang
%J American Journal of Computational Mathematics
%P 52-55
%@ 2161-1211
%D 2013
%I Scientific Research Publishing
%R 10.4236/ajcm.2013.33B009
%X <p class=\"MsoNormal\">
	<span lang=\"EN-US\">The planar Ramsey number <i>PR</i></span><i><span lang=\"EN-US\"> </span></i><span lang=\"EN-US\">(<i>H</i><sub>1</sub>, <i>H</i><sub>2</sub>) is the smallest integer <i>n</i> such that any planar graph on <i>n</i> vertices contains a copy of <i>H</i><sub>1</sub> or its complement contains a copy of <i>H</i><sub>2</sub>.
It is known that the Ramsey number <i>R</i>(<i>K</i><sub>4 </sub></span><span lang=\"X-NONE\">-</span><i><span lang=\"EN-US\">e</span></i><span lang=\"EN-US\">, <i>K</i><sub>6</sub>)
= 21, and the planar Ramsey numbers <i>PR</i>(<i>K</i><sub>4 </sub></span><span lang=\"X-NONE\">- </span><i><span lang=\"EN-US\">e</span></i><span lang=\"EN-US\">, <i>K<sub>l</sub></i>)
for <i>l</i> ¡Ü 5 are known. In this paper,
we give the lower bounds on <i>PR</i></span><i><span lang=\"EN-US\"> </span></i><span lang=\"EN-US\">(<i>K</i><sub>4 </sub></span><span lang=\"X-NONE\">? </span><i><span lang=\"EN-US\">e</span></i><span lang=\"EN-US\">, <i>K<sub>l</sub></i>) and determine
the exact value of <i>PR</i></span><i><span lang=\"EN-US\"> </span></i><span lang=\"EN-US\">(<i>K</i><sub>4 </sub></span><span lang=\"X-NONE\">- </span><i><span lang=\"EN-US\">e</span></i><span lang=\"EN-US\">, <i>K</i><sub>6</sub>).</span> 
</p>
%K Planar Graph
%K Ramsey Number
%K Forbidden Subgraph
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=38531