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Series: Graph Theory Seminar

Grötzsch's theorem implies that every planar triangle-free graph is
3-colorable. It is natural to ask whether this can be improved. We prove that
every planar triangle-free graph on n vertices has fractional chromatic number
at most 3-1/(n+1/3), while Jones constructed planar triangle-free n-vertex
graphs with fractional chromatic number 3-3/(n+1). We also investigate additional
conditions under that triangle-free planar graphs have fractional chromatic
number smaller than 3-epsilon for some fixed epsilon > 0.(joint work with J.-S. Sereni and J. Volec)

Series: Graph Theory Seminar

Generalized triangle T_r is an r-graph with edges {1,2,…,r}, {1,2,…,r-1, r+1} and {r,r+1, r+2, …,2r-2}. The family \Sigma_r consists of all r-graphs with three edges D_1, D_2, D_3 such that |D_1\cap D_2|=r-1 and D_1\triangle D_2\subset D_3. In 1989 it was conjectured by Frankl and Furedi that ex(n,T_r) = ex(n,\Sigma_r) for large enough n, where ex(n,F) is the Tur\'{a}n function. The conjecture was proven to be true for r=3, 4 by Frankl, Furedi and Pikhurko respectively. We settle the conjecture for r=5,6 and show that extremal graphs are blow-ups of the unique (11, 5, 4) and (12, 6, 5) Steiner systems. The proof is based on a technique for deriving exact results for the Tur\'{a}n function from “local stability" results, which has other applications. This is joint work with Sergey Norin.

Series: Graph Theory Seminar

Suppose that each node of a rooted tree has a message that it wants to
pass up the tree to the root. How can we design a protocol that guarantees
all messages (eventually) reach there without being interfered with by
other messages, if the nodes themselves do not know the underlying
structure of the tree, or even whether their previous messages were
successfully transmitted or not? I will describe (near optimal) answers to
several variations of this problem, based on joint work with Marek Chrobak
(UCR), Laszek Gasieniec (Liverpool) and Dariusz Kowalski (Liverpool).

Series: Graph Theory Seminar

(Alon-Tarsi Conjecture): For even n, the number of even nxn Latin squares
differs from the number of odd nxn Latin squares.
(Stones-Wanless, Kotlar Conjecture): For all n, the number of even nxn
Latin squares with the identity permutation as first row and first column
differs from the number of odd nxn Latin squares of this type.
(Aharoni-Berger Conjecture): Let M and N be two matroids on the same
vertex set, and let A1,...,An be sets
of size n + 1 belonging to both M and N. Then there exists a set belonging
to both M and N and meeting all Ai.
We prove equivalence of the first two conjectures and a special case
of the third one and use these results to show that Alon-Tarsi Conjecture
implies Rota's bases conjecture for odd n and
any system of n non-singular real valued matrices where one of them is
non-negative and the remaining have non-negative inverses.Joint work with Ron Aharoni.

Series: Graph Theory Seminar

In this talk, we discuss our recent progress on the famous directed cycle
double cover conjecture of Jaeger. This conjecture asserts that every
2-connected graph admits a collection of cycles such that each edge is in
exactly two cycles of the collection. In addition, it must be possible to
prescribe an orientation to each cycle so that each edge is traversed in
both ways.
We plan to define the class of weakly robust trigraphs and prove that a
connectivity augmentation conjecture for this class implies general
directed cycle double cover conjecture.
This is joint work with Martin Loebl.

Series: Graph Theory Seminar

This talk is a sequel to the speaker's previous lecture given in
the January 31st Combinatorics Seminar, but attendance at the first talk is
not assumed. We begin by carefully reviewing our generalized cycle-cocyle
reversal system for partial graph orientations. A self contained
description of Baker and Norin's Riemann-Roch formula for graphs is given
using their original chip-firing language. We then explain how to
reinterpret and reprove this theorem using partial graph orientations. In
passing, the Baker-Norin rank of a partial orientation is shown to be one
less than the minimum number of directed paths which need to be reversed in
the generalized cycle-cocycle reversal system to produce an acyclic partial
orientation. We conclude with an overview of how these results extend to
the continuous setting of metric graphs (abstract tropical curves).

Series: Graph Theory Seminar

Consider a graph G and a specified subset A of vertices. An A-path is a path with both ends in A
and no internal vertex in A. Gallai showed that there exists a min-max formula for the maximum number of pairwise disjoint
A-paths. More recent work has extended this result, considering disjoint A-paths which satisfy various additional properties.
We consider the following model. We are given a list of {(s_i, t_i): 0< i < k} of pairs of vertices in A, consider
the question of whether there exist many pairwise disjoint A-paths P_1,..., P_t such that for all j,
the ends of P_j are equal to s_i and t_i for some value i. This generalizes the disjoint paths problem and is NP-hard
if k is not fixed. Thus, we cannot hope for an exact min-max theorem. We further restrict the question, and ask if there
either exist t pairwise disjoint such A-paths or alternatively, a bounded set of f(t) vertices intersecting all such paths. In
general, there exist examples where no such function f(t) exists; we present an exact characterization of
when such a function exists.
This is joint work with Daniel Marx.

Series: Graph Theory Seminar

An r-unform n-vertex hypergraph H is said to have the
Manickam-Miklos-Singhi (MMS) property if for every assignment of
weights to
its vertices with nonnegative sum, the number of edges whose total
weight
is nonnegative is at least the minimum degree of H. In this talk I will
show that for n>10r^3, every r-uniform n-vertex hypergraph with equal
codegrees has the MMS property, and the bound on n is essentially
tight up
to a constant factor. An immediate corollary of this result is the
vector
space Manickam-Miklos-Singhi conjecture which states that for n>=4k
and any
weighting on the 1-dimensional subspaces of F_q^n with nonnegative
sum, the
number of nonnegative k-dimensional subspaces is at least ${n-1 \brack
k-1}_q$. I will also discuss two additional generalizations, which
can be
regarded as analogues of the Erdos-Ko-Rado theorem on k-intersecting
families. This is joint work with Benny Sudakov.

Series: Graph Theory Seminar

By the 4-color theorem, every planar graph on n vertices has
an independent set of size at least n/4. Finding a simple
proof of this fact is a long-standing open problem.
Furthermore, no polynomial-time algorithm to decide whether
a planar graph has an independent set of size at least (n+1)/4
is known.
We study the analogous problem for triangle-free planar graphs.
By Grotzsch' theorem, each such graph on n vertices has an
independent set of size at least n/3, and this can be easily
improved to a tight bound of (n+1)/3. We show that for every k,
a triangle-free planar graph of sufficiently large tree-width has
an independent set of size at least (n+k)/3, thus giving a polynomial-time
algorithm to decide the existence of such a set. Furthermore,
we show that there exists a constant c < 3 such that every planar graph
of girth at least five has an independent set of size at least n/c.Joint work with Matthias Mnich.

Series: Graph Theory Seminar

While Robertson and Seymour showed that graphs are
well-quasi-ordered under the minor relation, it is well known
that directed graphs are not. We will present an exact
characterization of the minor-closed sets of directed graphs
which are well-quasi-ordered. This is joint work with M.
Chudnovsky, S. Oum, I. Muzi, and P. Seymour.