20 Dicembre, 2016 15:00 oclock

Sezione di Geometria, Algebra e loro applicazioni

Open problems on 2-factorizations and new powerful methods to attack them

Tommaso Traetta, Department of Mathematics, Ryerson University

Aula Seminari Terzo Piano

A 2-factorization of a graph G is a set F of spanning 2-regular subgraphs (i.e., 2-factors) whose edge-sets partition the edge-set of G. It is well known that G has a 2-factorization if and only if it is regular of even degree. However, if we specify t 2-factors, say F1, F2, ..., Ft, and ask for the factorization F to contain ?_i factors isomorphic to Fi, then the problem becomes much harder. When t = 1
we have the well-known Oberwolfach problem, attributed to G. Ringel who posed it in 1956. The case t = 2 is known as the Hamilton-Waterloo problem, whereas for greater values of t we speak of the generalized Oberwolfach problem. The case where t ? {1, 2} and G is the complete (equipartite) graph is the most studied one. Nonetheless, these problems are still open although they have received much attention lately.

In this talk I will emphasize the connection between 2-factorizations and sharply transitive sets of permutations. Then, I will focus on some of the most recent results and their powerful algebraic methods.

This seminar is organized within the PRIN 2012 Research project «Geometric Structures, Combinatorics and their Applications» Grant Registration number 2012XZE22K, funded by MIUR - Project coordinator Prof.ssa Norma Zagaglia