Joseph Malkevitch

I would like to make a few remarks about Branko's (and to a lesser extent Vic's) influence on my mathematical career.

I was aware early in my career as a math major as an undergraduate - and this was reinforced when I became a graduate student - that the modest mathematical talents I had were concentrated in the area of geometry. I was fortunate in my career as a graduate student at the University of Wisconsin in Madison to find a kindred spirit in Donald Crowe, the only member of the UW Faculty to call himself a geometer at the time I was there. At that point Don was interested in finite planes and some of my earliest research was on finite hyperbolic planes. However, it became clear to me quickly that finite planes are in many respects more a branch of algebra than of geometry.

So it was my good fortune to come across the AMS Volume 7, 1963, Proceedings of a Symposium in Pure Mathematics, Convexity, edited by Vic. This volume had many articles by Vic and Branko.  In particular, I read Branko's article on Measures of Symmetry for Convex Sets and his paper with Motzkin on Polyhedral Graphs. This work inspired my first "original work." Initially this concerned questions about area and perimeter bisectors. I noticed that Branko was giving a talk at an  AMS meeting in Chicago. I think it was in Jan. 1966. I went to that meeting and heard his talk (or met him at someone else's) and spoke to him briefly.

The Convexity volume led me to Grünbaum and Motzkin's paper in the Canadian Journal which dealt with "left-right" paths in 3-valent graphs. Based on this paper I came up with some new questions about left-right paths and other "coded" paths on polyhedra. I got increasingly interested in the connection between graph theory and geometrical ideas.

When Don Crowe left Wisconsin for 6th months to spend some time in Hungary, before he left I asked if he would have a problem if I approached Branko about checking the status of the ideas I had based on his papers with Grünbaum and Motzkin.  I was sophisticated enough to know by then that often people knew a lot more than appeared in their published papers in areas that were being actively pursued.  So I wrote Branko (this was the age before email - when I say I wrote, I meant a letter!) and asked if I could visit him in Seattle in the summer of 1967.

I did not get a response for a while but in about March of 1967 I heard back that he had not answered sooner because he was not sure of his own summer plans. He invited me to come and visit.

So I came to Seattle in the summer of 1967 to visit and work with Branko. In fact the first person I met when I arrived was Vic. Being used to the motley crew at UW he resembled more the look of an insurance salesman than a mathematician. Vic was very welcoming.  He took me into his office and offered me copies of his reprints.  He steered me away from one pile that was taller than the others and with a smile told me that that paper had an error in it. Vic also introduced me to David Barnette, David Larman, and Joe Zaks, and I think it was then that I first met them.

A few days later I met Branko for the first time. He was extremely generous with his time and his ideas and encouragement. We met regularly, about once a week, for over a month. One of the problems I arrived with was that I had noticed the phenomenon of what is today called Barnette's Conjecture (but almost certainly also known to Tutte), which states that a planar 3-connected 3-valent bipartite graph has a hamiltonian circuit.

I spent most of the summer working on Barnette's conjecture (in particular, I looked at the special case that involved multi-4-gons, now shown to be equivalent to the full conjecture) and on my conjectures that if one has a 3-valent plane 3-connected graph with all its faces multiples of 4 or 5, then the graph has "simple" left-right paths.

At the end of the summer I have proved nothing. This clearly saddened Branko, who was rooting for me to make progress and solve problems that could be used in my dissertation. He advised me to work on the 4-gon version extension of his work with Motzkin. I remember feeling in my gut that when I looked at his problem the right way, all three cases, including Grünbaum and Motzkin's theorem would "fall out."

Luckily for me, a few months after my return to Madison, on a day that I remember vividly for having a headache, something that happened to me rarely, I was drawing some diagrams, more precisely doodling, and looked down at my doodle and there was the proof of all three cases at once!

So I went into overdrive and by August of 1968, and I had written up my results, which Branko kindly vetted in addition to what Don Crowe had helped with, and I was launched on my career as a mathematician.

What those weeks working with Branko, and looking at Chapter 13 of Convex Polytopes had done for me was to get me steeped in the tools of using graph theory arguments - Steinitz's Theorem - in pursuit of combinatorial results about 3-polytopes.

Beyond extending in Branko's work exceptional faces in 3-polytopes, and coded paths in 3-polytopal graphs, I also had a chapter in my thesis related to Eberhard's Theorem.  Branko and Motzkin had shown that what today are called fullerenes exist for any positive integer number of hexagons except 1. I was able to show that the proof of Branko and Motzkin which used several cases could be carried out using a construction where the number of hexagons changed one at a time for the family of fullerenes. In a sense nearly all of my subsequent work has involved finding "Eberhard"-like results in additional contexts beyond those that Branko pioneered.

Thank you, Branko, for helping making all this possible!
Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451

Phone: 718-262-2550 (Voicemail available)

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