Fixed points

[koh]

     Dear Seqfans

    I counted number of fixed points of mappings which are related with a coloring problem.

    The definition of N_{m,n} is described in the summary of "7 color theorem" which is written at the end of the mail.
    It is a mapping G_n -> H_n.
    
    [Number of fixed points]

    Fixed points of N_{2,1} 

    o

    o o    o_o

    o o o    o o_o    o_o
                      |/
                      o
 
    o o o o    o_o o o    o_o o_o        

    o_o o    o_o
    |/       |x|
    o        o_o

    "Number of fixed points" are as follows.
    S1 : 1,2,3,5....

    This is the same as A000041.


    Fixed points which are connected graphs of N_{2,2}

    o

    o_o

    o_o_o    o_o
             |/
             o

    o_o_o_o    o_o_o    o_o_o    o_o
                 |      |/       |x|
                 o      o        o_o 

    o_o_o_o_o    o_o_o_o    o_o_o_o    o_o_o      o_o_o_o    
                   |        |/         |   |        |/  
                   o        o          o___o        o

         o_o_o     o_o_o     o_o_o      o_o_o    o_o_o
          /|       |/|       |/         |x|      |x-x|
         o o       o o       o_o        o_o      o___o

    These graohs don't have the subgraphs as follows.

         o_o    o_o
         | |    |/|
         o_o    o_o

    The tenth five nodes graph is K_5.

    "Number of connected fixed points" are as follows.
    S2 : 1,1,2,4,10....

    To Neil : 
    Don't submit the sequence until when I will have edited it.
    Recently you submitted my sequences too quickly.
    

    [Summary of 7 color  theorem]

    We consider about coloring problem, so no loop exists in a graph.
    If multiple edges exist then remove them.

    For all different two vertices x,y in G if n paths of length m exist between x and y then add an edge xy. 
    The graph H which is made from G is represented as N_{m,n}(G).

    N_{m,n} is a mapping form k nodes graph to k nodes graph.

          N is for "Near"
 
    [Example}

        N_{1,1}(G)=G

    [Other definition of N_{2,3}]

    G={V,E_g}, H=N_{2,3}(G), H={V,E_h}

    All x,y (x,y E V and -x=y and (Exist p,q,r -p=q and -p=r and -q=r and xp,xq,xr,py,qy,ry E E_g)) -> E_h = E_g U {xy}

    Where "E" means "element". "-" means "not"

             x          x 
            /| |        |
           p q r   +    |   
            V /         |
            y           y

          3 paths 
          xpy xqy xry
          exist
          then
          add xy
          
    [T.1]

    Chr(N_{2,1}(G))=infinity

         Where Chr means chromatic number. G are planer graph. 

    Proof : 
    Consider the following graph G_n.
    G_n={V,E}, V={p,x_i}, E={px_i}, 1<=i<=n
    For all i,j one path x_ipx_j whose length is 2 exists, so N_{2,1}(G_n) becomes K_{n+1} and Chr(K_{n+1})=n+1, then limit_{n -> infinity} Chr(K_{n+1})=infinity

    [T.2}
  
    Chr(N_{2,2}(G))=infinity

    Proof : 
    Consider the following graph G_n.
    G_n={V,E}, V={p,x_i,q}, E={px_i,x_iq}, 1<=i<=n
         
    [T.7C]

    Chr(N_{2,3}(G))=7

    Proof : 
    No difficulty like "four color theorem" exists but it is complicated.

    Yasutoshi