On a Unimodality Conjecture in Matroid Theory
نویسنده
چکیده
A certain unimodal conjecture in matroid theory states the number of rank-r matroids on a set of size n is unimodal in r and attains its maximum at r = bn/2c. We show that this conjecture holds up to r = 3 by constructing a map from a class of rank-2 matroids into the class of loopless rank-3 matroids. Similar inequalities are proven for the number of non-isomorphic loopless matroids, loopless matroids and matroids.
منابع مشابه
Independence Sequences of Well-Covered Graphs: Non-Unimodality and the Roller-Coaster Conjecture
A graph G is well-covered provided each maximal independent set of vertices has the same cardinality. The term sk of the independence sequence (s0, s1, . . . , sα) equals the number of independent k-sets of vertices of G. We investigate constraints on the linear orderings of the terms of the independence sequence of well-covered graphs. In particular, we provide a counterexample to the recent u...
متن کاملCyclic orderings and cyclic arboricity of matroids
We derive a general result concerning cyclic orderings of the elements of a matroid. As corollaries we obtain two further results. The first corollary proves a conjecture of Gonçalves [7], stating that the circular arboricity of a matroid is equal to its fractional arboricity. This generalises a well-known result from Nash-Williams on covering graphs by spanning trees, and a result from Edmonds...
متن کاملThe Chip Firing Game and Matroid Complexes
In this paper we construct from a cographic matroid M , a pure multicomplex whose degree sequence is the h–vector of the the matroid complex of M. This result proves a conjecture of Richard Stanley [Sta96] in the particular case of cographic matroids. We also prove that the multicomplexes constructed are M–shellable, so proving a conjecture of Manoj Chari [Cha97] again in the case of cographic ...
متن کاملA Simple Proof of a Conjecture of Simion
Simion had a unimodality conjecture concerning the number of lattice paths in a rectangular grid with the Ferrers diagram of a partition removed. Hildebrand recently showed the stronger result that these numbers are log concave. Here we present a simple proof of Hildebrand’s result.
متن کاملNowhere-zero Flows in Regular Matroids and Hadwiger’s Conjecture
We present a tool that shows, that the existence of a k-nowhere-zero-flow is compatible with 1-,2and 3-sums in regular matroids. As application we present a conjecture for regular matroids that is equivalent to Hadwiger’s conjecture for graphs and Tuttes’s 4and 5-flow conjectures.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Discrete Mathematics & Theoretical Computer Science
دوره 5 شماره
صفحات -
تاریخ انتشار 2002