A350716 - OEIS

%I #29 Dec 01 2023 05:22:16

%S 4,4,4,4,4,4,4,5,5,6,6,6,7,7,8,8,8,9,9,10,10,10,11,11,12,12,12,13,13,

%T 14,14,14,15,15,16,16,16,17,17,18,18,18,19,19,20,20,20,21,21,22,22,22,

%U 23,23,24,24,24,25,25,26,26,26,27,27,28,28,28,29,29,30

%N a(n) is the minimum number of vertices of degree 3 over all 3-collapsible graphs with n vertices.

%C A graph G is k-collapsible if it has minimum degree k and has no proper induced subgraph with minimum degree k.

%H Paolo Xausa, <a href="/A350716/b350716.txt">Table of n, a(n) for n = 4..10000</a>

%H Allan Bickle, <a href="https://allanbickle.files.wordpress.com/2016/05/dissertation4.pdf">The k-Cores of a Graph</a>, Ph.D. Dissertation, Western Michigan University (2010).

%H Allan Bickle, <a href="https://allanbickle.files.wordpress.com/2016/05/collapsiblepaper.pdf">Collapsible graphs</a>, Congr. Numer. 231 (2018), 165-172.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).

%F a(n) = ceiling(2*n/5) = A057354(n) for n > 7.

%F G.f.: x^4*(4 - 4*x^5 + x^7 + x^9)/((x^4 + x^3 + x^2 + x + 1)*(x - 1)^2). - _Stefano Spezia_, Feb 05 2022

%e For n between 4 and 6, 3-collapsible graphs with 4 degree 3 vertices are:

%e - a complete graph with 4 vertices,

%e - a wheel with 5 vertices,

%e - the graph formed by removing a 4-cycle and a 2-clique from a complete graph with 6 vertices.

%t A350716[n_]:=If[n<8,4,Ceiling[2n/5]];

%t Array[A350716,100,4] (* _Paolo Xausa_, Dec 01 2023 *)

%o (Python)

%o print([4,4,4,4] + [2*n//5 for n in range(10, 80)]) # _Gennady Eremin_, Feb 05 2022

%Y Cf. A351169, A057354.

%K nonn,easy

%O 4,1

%A _Allan Bickle_, Feb 03 2022