1,924 research outputs found
Proof of a tournament partition conjecture and an application to 1-factors with prescribed cycle lengths
In 1982 Thomassen asked whether there exists an integer f(k,t) such that
every strongly f(k,t)-connected tournament T admits a partition of its vertex
set into t vertex classes V_1,...,V_t such that for all i the subtournament
T[V_i] induced on T by V_i is strongly k-connected. Our main result implies an
affirmative answer to this question. In particular we show that f(k,t) = O(k^7
t^4) suffices. As another application of our main result we give an affirmative
answer to a question of Song as to whether, for any integer t, there exists an
integer h(t) such that every strongly h(t)-connected tournament has a 1-factor
consisting of t vertex-disjoint cycles of prescribed lengths. We show that h(t)
= O(t^5) suffices.Comment: final version, to appear in Combinatoric
On realization graphs of degree sequences
Given the degree sequence of a graph, the realization graph of is the
graph having as its vertices the labeled realizations of , with two vertices
adjacent if one realization may be obtained from the other via an
edge-switching operation. We describe a connection between Cartesian products
in realization graphs and the canonical decomposition of degree sequences
described by R.I. Tyshkevich and others. As applications, we characterize the
degree sequences whose realization graphs are triangle-free graphs or
hypercubes.Comment: 10 pages, 5 figure
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