### In Memoriam Roger Charles Entringer

Dr Entringer earned his Ph.D. at the University of New Mexico in 1963 with Dissertation: Some Properties of Certain Sets of Coprime Integers under Joerg Werner von Peter (Kalkschmidt) Mayer.  He was Professor Emeritus at the University of New Mexico, having supervised six students during his time there, Curtis Barefoot, Joseph Barr, Lane Clark, Charles Harner, Joe McCanna, and Tom Porter.

Swart spent a sabbatical leave at the University of New Mexico, Albuquerque, New Mexico, USA. She attended the graph theory lectures delivered by Roger Charles Entringer in which he handed out a set of open questions to the class. Swart thought that these were exercises for the class and attempted to solve them but was surprised to find that she could not solve some of them. Entringer was, however, very impressed with the progress that she had made and this led to a joint publication by Swart and Entringer, Spanning cycles of nearly cubic graphs (1980). The date on this paper is slightly confusing since the two authors submitted it for publication in July 1977 while Swart was at the University of New Mexico. The paper was reviewed by A G Thomason who wrote:-
The fact that a cubic hamiltonian graph must have at least three spanning cycles suggests the question of whether every hamiltonian graph in which each point has degree at least 3 must have at least three spanning cycles. We answer this in the negative by exhibiting graphs on n = 2m + 1, m > 5, points in which one point has degree 4, all others have degree 3, and only two spanning cycles exist. A cubic Hamilton graph has at least three Hamilton cycles. The authors construct graphs having exactly two Hamilton cycles, in which every vertex except one has degree 3, and one vertex has degree 4. They also exhibit graphs with two vertices of degree 4, all others of degree 3, and possessing exactly one Hamilton cycle.

Notably, he is credited with the Entringer numbers E(n,k), the number of permutations of {1, 2, …, n+1},, starting with k+1, which, after initially falling, alternately fall then rise. The Entringer numbers are given by E(0,0)=1, E(n,0)=0 together with the recurrence relation E(n,k)=E(n,k-1)+E(n-1,n-k).

Gus Simmons reports that Rodger was quite proud that he was able to coauthor three papers with Paul Erdős:

L. Clark, R. Entringer, P. Erdős, H.C. Sun, L. Székely, Extremal problems for the Bondy-Chvátal closure of a graph, Graphs, matrices, and designs, pages 73–83 in Lecture Notes in Pure and Appl. Math., 139, Dekker, New York, 1993.

R.C. Entringer, P. Erdős an C.C. Harner, Some extremal properties concerning transitivity in graphs, Period. Math. Hungar. 3(3-4) (1973), 275–279.

R.C. Entringer, Paul Erdős, On the number of unique subgraphs of a graph, J. Combinatorial Theory Ser. B 13 (1972), 112–115.

One of the open problems Erdős liked to include in his “open problems” lectures was: What is the length of the longest arithmetic progression whose appearance can’t be avoided in a singly (doubly) infinite rearrangement of the positive integers? Roger, Jim Davis, Ron Graham and Gus solved
Erdős’s problem for all but a single case!

J.A. Davis, R.C. Entringer, R.L Graham and G.J. Simmons, On permutations containing no long arithmetic progressions, Acta Arith. 34(1) (1977/78), 81–90.

Some time ago Roger called Gus to say; We haven’t been forgotten. Someone just solved the missing case.

H. Ardal, T. Brown and V. Jungić, Chaotic orderings of the rationals and reals, Amer. Math.      Monthly 118(10) (2011), 921–925.

1. Roger Entringer's 42 years "Prime Tree Graph Conjecture", now solved and send to Journal

1. Any one comments on Entringer Prime Tree Conjecture