Combinatorics and Graph Theory

2511 Submissions

[2] viXra:2511.0120 [pdf] submitted on 2025-11-23 09:10:25

A Note on the Occurence of Fibonacci Terms in the Differences Between n and Pn of Pancake Graphs

Authors: Ryan O'Rourke
Comments: 12 Pages.

In this paper I discuss a pattern observable in the n and Pn values of the first nineteen pancake graphs. This pattern could potentially hint at the as yet unknown diameters of the next graphs, and is at least viable for first seventy-four values of n, since it predicts values, in the aforementioned range, of Pn which fall between published lower and upper bounds. The pattern arises in sets of adjacent n's with equal differences between n and Pn, and is equivalent to a subsequence of the Fibonacci sequence. If one takes the known values of Pn and deletes n from each, one gets a difference value, which we call d, which allows one to arrange the numbers into corresponding blocks, so that the first block has 2 columns, the second 3, then 5 and then 8, and there a Fibonacci subsqeuence appears to be emerging (...2,3,5,8...). In this paper, I provide a formula for Pn for those n that follow this pattern:h(n) = n + ⌈(logφ (-(n - 1)(√5 - √5φ) + φ3)) - 4⌉ - 1 , and test it against published upper and lower bounds for Pn for n ≤ 10000.
Category: Combinatorics and Graph Theory

[1] viXra:2511.0046 [pdf] submitted on 2025-11-11 20:10:13

Non-Best Bounds for Schur Numbers Enjoying Simple Proofs

Authors: Warren D. Smith
Comments: 4 Pages.

The Kth "Schur number" S(K) is the least positive integer N such that for every coloring of the integers {1,2,3,...,N} with K colors, an equation a+b=c exists with 1≤a≤b≤c≤N with a,b,c all having the same color. Equivalently it is the greatest possible N so that a (K-1)-coloring of {1,2,3,...,N-1} exists avoiding monochromatic a+b=c. [The latter formulation has the advantage of telling us that S(0)=1.] We give very simple proofs that GK≤S(K)≤⌈K!(e-1/24)⌉ where e≈2.71828 for various growth-factors G=3/2, G=2, and (for k≥5) G=1611/5=2.76290... These Gs are not the best known, but my proofs are very simple. We speculate that S(K) grows superexponentially, and (if that is true) explain how a computer could prove arbitrarily large Gs or (if it is not true) prove a G arbitrarily near to maximum possible, after enough work. I have not been able to prove superexponentiality, but the final paragraph explains a strategy that might be able to.
Category: Combinatorics and Graph Theory