

A223887


Number of 4colored labeled graphs on n vertices.


9



1, 4, 28, 340, 7108, 254404, 15531268, 1613235460, 284556079108, 85107970698244, 43112647751430148, 36955277740855136260, 53562598422461559373828, 131186989945696839128432644, 542676256323680030599454982148
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OFFSET

0,2


COMMENTS

A simple graph G is a kcolorable graph if it is possible to assign one of k' <= k colors to each vertex of G so that no two adjacent vertices receive the same color. Such an assignment of colors is called a coloring function for the graph.
A kcolored graph is a kcolorable graph together with its coloring function. This sequence gives the number of labeled 4colored graphs on n vertices. An example is given below.
See A047863 for labeled 2colored graphs on n vertices and A191371 for labeled 3colored graphs on n vertices. See A076316 for labeled 4colorable graphs on n vertices and A224068 for the count of labeled graphs colored using exactly 4 colors.


REFERENCES

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973.


LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..50
S. R. Finch, Bipartite, kcolorable and kcolored graphs
S. R. Finch, Bipartite, kcolorable and kcolored graphs, June 5, 2003. [Cached copy, with permission of the author]
R. C. Read, The number of kcolored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410—414.
R. P. Stanley, Acyclic orientation of graphs Discrete Math. 5 (1973), 171178. North Holland Publishing Company.
Eric Weisstein's World of Mathematics, kColorable Graph


FORMULA

a(n) = sum {k = 0..n} binomial(n,k)*2^(k*(nk))*b(k)*b(nk), where b(n) := sum {k = 0..n} binomial(n,k)*2^(k*(nk)).
Let E(x) = sum {n >= 0} x^n/(n!*2^C(n,2)). Then a generating function for this sequence is E(x)^4 = sum {n >= 0} a(n)*x^n/(n!*2^C(n,2)) = 1 + 4*x + 28*x^2/(2!*2) + 340*x^3/(3!*2^3) + .... In general, for k = 1, 2, ..., E(x)^k is a generating function for labeled kcolored graphs (see Stanley).


EXAMPLE

a(2) = 28: There are two labeled 4colorable graphs on 2 nodes, namely
A) 1 2 B) 1 2
o o oo
Using 4 colors there are 16 ways to color the graph of type A and 4*3 = 12 ways to color the graph of type B so that adjacent vertices do not share the same color. Thus there are in total 28 labeled 4colored graphs on 2 vertices.


PROG

(PARI)
N=66; x='x+O('x^N);
E=sum(n=0, N, x^n/(n!*2^binomial(n, 2)) );
tgf=E^4; v=concat(Vec(tgf));
v=vector(#v, n, v[n] * (n1)! * 2^((n1)*(n2)/2) )
/* Joerg Arndt, Apr 10 2013 */


CROSSREFS

Column k=4 of A322280.
Equals 4 * A000686, A047863 (labeled 2colored graphs), A076316, A191371 (labeled 3colored graphs), A224068.
Sequence in context: A113371 A280570 A080636 * A140425 A193198 A165193
Adjacent sequences: A223884 A223885 A223886 * A223888 A223889 A223890


KEYWORD

nonn,easy


AUTHOR

Peter Bala, Apr 10 2013


STATUS

approved



