<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML><HEAD>
<META http-equiv=Content-Type content="text/html; charset=iso-2022-jp">
<META content="MSHTML 6.00.2900.2180" name=GENERATOR>
<STYLE></STYLE>
</HEAD>
<BODY bgColor=#ffffff>
<DIV><FONT face="MS UI Gothic">
<DIV><FONT face="MS UI Gothic"> Hello,
seqfans. </FONT></DIV>
<DIV><FONT
face="MS UI Gothic">
</FONT><FONT
face="MS UI Gothic"> <BR> </FONT><FONT
face="MS UI Gothic"> This is a sequence from OEUAI.<BR>
ID Number : B020004 <BR> Sequence : 7, 12, <BR>
<BR> Name : Chromatic number of Near_{2,3}(G_k). <BR>
Where G_k is graphs on a surface with k holes. <BR></FONT></DIV>
<DIV><FONT face="MS UI Gothic"> A mapping Near_{n,m} : H ->
H' is defined as follows. <BR> If we add all edges
between two vertices which are (m,n) on H, then H becomes Near_{m,n}(H).
<BR> Definition of (m,n) : <BR> n edges of
length m exist between different two
vertices. <BR>
Example :
<BR>
If V={x,y,a,b,c} , E={xa, xb, xc, ay, by, cy} , then x,y is (2,3).
<BR>
If a graph is like this o-o-o , then the virtices of both sides is
(2,1). </FONT></DIV>
<DIV></FONT><FONT face="MS UI Gothic"><FONT face="MS UI Gothic"><FONT
size=2></FONT><BR> Comment : a(0)=1 is
proved. a(1)=12 is lower
bound.<BR>
I am not sure at all that the Heawood like formula is
correct. </FONT></DIV></DIV>
<DIV>
<DIV> </DIV>
<DIV>
Kai(Near_{2,3}(G(E)) = [(13 + (169 -
48*E)^(1/2))/2] </DIV>
<DIV>
E 0, 1, 2,
3, 4, 5 </DIV>
<DIV>
13, 14, 15, 17, 18, 19</DIV>
<DIV><FONT face="MS UI Gothic" size=2>
<DIV><FONT face="MS UI Gothic"
size=2> <FONT
size=3> where G(E) is a graph which has
Euler number E, E<=0</FONT></FONT></DIV></FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT
face="MS UI Gothic">
I dont'n remember the reason why I made it.<BR></FONT></DIV>
<DIV><FONT face="MS UI Gothic"> Keyword : unknown <BR> Offset : 1 <BR>
Author : Yasutoshi <BR> <BR></FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> To know the Chromatic number
of Near_{2,3}(G_k) is rather dificult.<BR> So, I considered a
little more easy .</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> %S A000001 7, 12,
<BR> %N A000001 a(k)=Minimal number of vertices of a S_k
graph which satisfies the following condition. </FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT
face="MS UI Gothic">
C: For all two different vertices on a
graph,<BR>
an edge exists between them
<BR>
or,
<BR>
three paths of length two exist between them.</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT
face="MS UI Gothic">
Where S_k means a surface with k holes.</FONT></DIV>
<DIV><FONT face="MS UI Gothic"> <BR> %C A000001 The
formula might give a upper bound.<BR> <BR>
For example, if we search the existance of a graph with 14 nodes which satisfies
the condition using a computer, the number of all graphs is 2^C_{14,2} = 2^91=
30 digits<BR> The digits is too large to do exhaustive
search, but if some good algorithm exists, it might possible to find the
graph. <BR> </FONT></DIV>
<DIV><FONT face="MS UI Gothic"> Yasutoshi</FONT></DIV>
<DIV><FONT face="MS UI Gothic">
<BR></DIV></FONT></FONT></DIV></BODY></HTML>