Alison Marr - cp's OEIS frontend

User: Alison Marr

Alison Marr has authored 3 sequences.

A227299 Number of vertices for which there is an alien in-magic directed star.

6, 10, 14, 15, 18, 21, 28, 33, 36, 39, 45, 55, 60, 66, 68, 78, 91, 95, 105, 120, 136, 138, 150, 153, 171, 189, 190, 203, 210, 231, 248, 253, 264, 276, 300, 315, 325, 333, 351, 378, 390, 406, 410, 435, 465, 473, 495, 496, 528, 561, 564, 588, 595, 630, 663, 666, 689, 703, 741, 770, 780, 798, 820, 861, 885

Offset: 1

Alison Marr, Jul 11 2013

nonn

Numbers v such that there is an e in (v+1,2v-2) with s=(v+e)(v+e+1)/2v being an integer and c ceiling((2(v+e)-s)/2)=v-1.
This implies there exist an in-magic labeling of a directed star (K_{1,v-1}) containing two or more doubled edges (where a doubled edge is one in which there are two edges between a pair of vertices but the orientations of the two edges are distinct).
A graph with v vertices and e edges has an in-magic labeling if we label both the vertices and integers with the numbers 1, 2, ..., v+e using each number exactly once so that at any vertex, the sum of the vertex plus the sum of the labels on the incoming edges equals the same value for each vertex in the graph.
The first condition in our result is equivalent to sequence A024619. Thus, this is a subsequence of A024619. (Note: There are two such e for v=105).

A213273 The smallest m such that the complete bipartite graph K_{n,n} has a coprime labeling using labels from {1,...,m}.

Original entry on oeis.org

2, 4, 7, 9, 11, 15, 17, 21, 23, 27, 29, 32, 37, 40, 43, 46, 49, 53, 57, 61, 63, 67, 71, 73, 77, 81, 83, 88, 92, 97, 100, 103, 107, 111, 113, 118, 122, 125, 128, 133, 135, 139, 143, 147, 149, 153, 157, 163, 165, 167, 171, 173, 178, 181, 188, 191, 194, 197, 202

Offset: 1

Adam Berliner, Nate Dean, Jonelle Hook, Alison Marr, Aba Mbirika, Cayla McBee, Jun 08 2012

nonn

A prime labeling of a graph G is a labeling of the vertices with the integers 1, 2, ..., v (where v is the number of vertices) such that any two adjacent vertices have labels that are relatively prime. Here we are allowing the largest label m >= v and calling that a coprime labeling. Our goal is to find the smallest m that makes the labeling possible for K_{n,n} (which clearly does not have a prime labeling for n>2).

			For n=12 and K_{12,12} the two independent sets would be labeled {1,3,5,9,15,17,19,23,25,27,29,31} and {2,4,7,8,11,13,14,16,22,26,28,32}.

Kevin Cuadrado, Table of n, a(n) for n = 1..2000 (terms 1..14 from Adam H. Berliner, N. Dean, J. Hook, A. Marr, A. Mbirika & C. McBee, terms 14..23 from Alois P. Heinz, terms 24..96 from Paul Tabatabai).
Adam H. Berliner, N. Dean, J. Hook, A. Marr, A. Mbirika, and C. McBee, Coprime and prime labelings of graphs, arXiv preprint arXiv:1604.07698 [math.CO], 2016, Coprime and Prime Labelings of Graphs, Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.8.
Gary Chartrand, Cooroo Egan, and Ping Zhang, "Harmonious Labelings", How to Label a Graph (2019), Springer Briefs in Mathematics, Springer, Cham, 21-28.
Catherine Lee, Minimum coprime graph labelings, arXiv:1907.12670 [math.CO], 2019.

Cf. A213806, A284875, A291465.

b:= proc(n, k, t, s) option remember;
      nops(s)>=t and (k>=t or n>1 and (b(n-1, k, t, s) or
      b(n-1, k+1, t, select(x-> igcd(n, x)=1, s))))
    end:
a:= proc(n) option remember; local m; forget(b);
      for m from `if`(n=1, 1, a(n-1))
      while not b(m, 1, n, {$2..m}) do od; m
    end:
seq(a(n), n=1..14);  # Alois P. Heinz, Jun 16 2012

b[n_, k_, t_, s_] := b[n, k, t, s] = Length[s] >= t && (k >= t || n > 1 && (b[n - 1, k, t, s] || b[n - 1, k + 1, t, Select[s, GCD[n, #] == 1 &]]));
a[n_] := a[n] = Module[{m}, m = If[n == 1, 1, a[n - 1]]; While[!b[m, 1, n, Range[2, m]], m++]; m];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 23}] (* Jean-François Alcover, Nov 06 2017, after Alois P. Heinz *)

More terms from Alois P. Heinz, Jun 16 2012

a(24) and beyond from Paul Tabatabai, Apr 29 2019

A105206 Number of edges in a pancyclic graph on n+2 vertices with the fewest possible edges.

Original entry on oeis.org

3, 5, 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26

Offset: 3

John C. George (jgeorge(AT)gdn.edu), Walter D. Wallis (wdwallis(AT)siu.edu) and Alison Marr, Apr 12 2005

A graph on n vertices is said to be pancyclic if there are cycles of each length 3, 4, ... n in the graph.

			For n = 3 the answer is 3; each of the three vertices is connected to each other vertex, forming a 3-cycle. For n = 4 we find it takes five edges and for n = 5 it takes 6.

Different from A080036.

a(14)-a(22) by Alison Marr, Aug 22 2011

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User: Alison Marr

A227299 Number of vertices for which there is an alien in-magic directed star.

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A213273 The smallest m such that the complete bipartite graph K_{n,n} has a coprime labeling using labels from {1,...,m}.

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A105206 Number of edges in a pancyclic graph on n+2 vertices with the fewest possible edges.

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