A342357 Number of fundamentally different rainbow graceful labelings of graphs with n edges.

1, 2, 11, 125, 1469, 30970, 1424807, 25646168, 943532049, 66190291008, 1883023236995, 119209289551407, 8338590851427689, 366451025462807402, 25231464507361789935, 2996947275258886238380, 211289282287835811874277, 12680220578500976681544666, 1815313698001596651227722787

Offset

1,2

Comments

Rainbow graceful labelings are also known as rho-labelings, as originally introduced by Rosa in 1967.

Equivalently, they are graceful labelings of the digraph obtained by replacing each edge by a pair of arcs in opposite directions.

Consider vertices numbered 0 to 2n. For 1 <= k <= n, add an edge between v_k and (v_k+k) mod q, where q = 2n+1. (Thus (2n+1)^n possibilities.) Two such graphs are considered equivalent under the following operations: (i) rename each v to (v+1) mod q; (ii) rename each v to (av) mod q, where a is relatively prime to q. The number of equivalence classes is a(n).

References

D. E. Knuth, The Art of Computer Programming, forthcoming exercise in Section 7.2.2.3.

A. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs (Internat. Symposium, Rome, July 1966), Dunod Paris (1967) 349-355.

Links

Peter Luschny, Table of n, a(n) for n = 1..100

R. Montgomery, A. Pokrovskiy, and B. Sudakov, A proof of Ringel's Conjecture, arXiv:2001.02665 [math.CO], 2020.

Example

Each equivalence class has exactly one graph with v_1=0.
For n=3 the eleven classes of graphs 0v_2v_3 are: {000,011,015,050,054,065}, {001,002,024,041,063,064}, {003,026,031,034,046,062}, {004,061}, {005,013,021,044,052,060}, {006,014,030,035,051,066}, {010,055}, {012,020,022,043,045,053}, {016,025,032,033,040,056}, {023,042}, {036}.

Mathematica

sols[alf_, bet_, q_]:=Block[{d=GCD[alf, q]}, If[Mod[bet, d]!=0, 0, d]]
(* that many solutions to alf x == bet (modulo q) for 0<=x<q *)
f[l_, a_, b_, q_]:=Block[{r, s, ll, atos},
   s=1; ll=Mod[l*a, q]; r=1;
   While[ll>l && q-ll>l, s++; ll=Mod[ll*a, q]; r=Mod[r*a+1, q]];
   If[ll==l, sols[a^s-1, -r b, q], If[q-ll==l, sols[a^s-1, l-r b, q], 1]]]
f[a_, b_, q_]:=Product[f[l, a, b, q], {l, (q-1)/2}]
x[q_]:=Sum[If[GCD[a, q]>1, 0, Sum[f[a, b, q], {b, 0, q-1}]], {a, q-1}]/(q EulerPhi[q])
a[n_]:=x[2n+1]

Prog

(SageMath) # This is a port of the Mathematica program.
def sols(a, b, q):
    g = gcd(a, q)
    return 0 if mod(b, g) != 0 else g
def F(k, a, b, q):
    s, r, m = 1, 1, mod(k*a, q)
    while m > k and q - m > k:
        s += 1
        m = mod(m*a, q)
        r = mod(r*a + 1, q)
    if m == k:   return sols(a^s - 1, -r*b, q)
    if m == q-k: return sols(a^s - 1, k - r*b, q)
    return 1
def f(a, b, q):
    return prod(F(k, a, b, q) for k in (1..(q-1)//2))
def a(n):
    q = 2*n + 1
    s = sum(0 if gcd(a, q) > 1 else sum(f(a, b, q)
        for b in (0..q-1)) for a in (1..q-1))
    return s // (q*euler_phi(q))
print([a(n) for n in (1..19)]) # Peter Luschny, Mar 10 2021

Crossrefs

Cf. A341884, A342225, A005193, A002618.

Sequence in context: A269069 A361036 A224366 * A279703 A206401 A193207

Adjacent sequences: A342354 A342355 A342356 * A342358 A342359 A342360

Keyword

nonn

Author

Don Knuth, Mar 09 2021

Status

approved