A223887 Number of 4-colored labeled graphs on n vertices.
1, 4, 28, 340, 7108, 254404, 15531268, 1613235460, 284556079108, 85107970698244, 43112647751430148, 36955277740855136260, 53562598422461559373828, 131186989945696839128432644, 542676256323680030599454982148
Offset: 0
Examples
a(2) = 28: There are two labeled 4-colorable graphs on 2 nodes, namely
A) 1 2 B) 1 2
o o o----o
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 4-colored graphs on 2 vertices.
References
- F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973.
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..50
- Steven R. Finch, Bipartite, k-colorable and k-colored graphs
- Steven R. Finch, Bipartite, k-colorable and k-colored graphs, June 5, 2003. [Cached copy, with permission of the author]
- R. C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410-414.
- R. P. Stanley, Acyclic orientation of graphs, Discrete Math. 5 (1973), 171-178. North Holland Publishing Company.
- Eric Weisstein's World of Mathematics, k-Colorable Graph
Crossrefs
Programs
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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] * (n-1)! * 2^((n-1)*(n-2)/2) ) /* Joerg Arndt, Apr 10 2013 */
Formula
a(n) = sum {k = 0..n} binomial(n,k)*2^(k*(n-k))*b(k)*b(n-k), where b(n) := sum {k = 0..n} binomial(n,k)*2^(k*(n-k)).
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 k-colored graphs (see Stanley).
Comments