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Series: Graph Theory Seminar

We will discuss the classic theorem of Walter Schnyder: a graph G is planar if and only if the dimension of its incidence poset is at most 3. This result has been extended by Brightwell and Trotter to show that the dimension of the vertex-edge-face poset of a planar 3-connected graph is 4 and the removal of any vertex (or by duality, any face) reduces the dimension to 3. Recently, this result and its extension to planar multigraphs was key to resolving the question of the dimension of the adjacency poset of a planar bipartite graph. It also serves to motivate questions about the dimension of posets with planar cover graphs.

Series: Graph Theory Seminar

Slightly modifying a quote of
Paul Erdos: The problem for graphs we
solve this week. The corresponding problem
for posets will take longer.
As one example, testing a graph to determine
if it is planar is linear in the number of
edges. Testing an order (Hasse) diagram to
determine if it is planar is NP-complete.
As a second example, it is NP-complete
to determine whether a graph is a cover
graph.
With these remarks in mind, we present
some results, mostly new but some classic,
regarding posets with planar cover graphs
and planar diagrams. The most recent
result is that for every h, there is a constant
c_h so that if P is a poset of height h and
the cover graph of P is planar, then
the dimension of P is at most c_h.

Series: Graph Theory Seminar

We study several parameters of cubic graphs with large girth. In particular, we prove that every n-vertex cubic graph with sufficiently large girth satisfies the following:

- has a dominating set of size at most 0.29987n (which improves the previous bound of 0.32122n of Rautenbach and Reed)
- has fractional chromatic number at most 2.37547 (which improves the previous bound of 2.66881 of Hatami and Zhu)
- has independent set of size at least 0.42097n (which improves the previous bound of 0.41391n of Shearer), and
- has fractional total chromatic number arbitrarily close to 4 (which answers in the affirmative a conjecture of Reed). More strongly, there exists g such that the fractional total chromatic number of every bridgeless graph with girth at least g is equal to 4.

The presentation is based on results obtained jointly with Tomas Kaiser, Andrew King, Petr Skoda and Jan Volec.

Series: Graph Theory Seminar

Richter and Salazar conjectured that graphs that are critical for a fixed crossing number k have bounded bandwidth. A weaker well-known conjecture of Richter is that their maximum degree is bounded in terms of k. We disprove these conjectures for every k >170, by providing examples of k-crossing-critical graphs with arbitrarily large maximum degree, and explore the structure of such graphs.

Series: Graph Theory Seminar

A well know theorem of Kuratowski states that a graph is planar graph iff it contains no TK_5 or TK_{3,3}. In 1970s Seymour conjectured that every 5-connected nonplanar graph contains a TK_5. In the talk we will discuss several special cases of the conjecture, for example the graphs containing K_4^- (K_4 withour an edge). A related independent paths theorem also will be covered.

Series: Graph Theory Seminar

I will discuss some new results, as well as new interpretations of some old results, concerning reduced divisors (a.k.a. G-parking functions) on graphs, metric graphs, and tropical curves.

Series: Graph Theory Seminar

Many results in asymptotic extremal combinatorics are obtained using just a handful of instruments, such as induction and Cauchy-Schwarz inequality. The ingenuity lies in combining these tools in just the right way. Recently,
Razborov developed a flag calculus which captures many of the available techniques in pure form, and allows one, in particular, to computerize the search for the right combination. In this talk we outline the general approach and describe its application to the conjecture that a digraph with minimum outdegree n/3 contains a directed triangle. This special case of the Caccetta-Haggkvist conjecture has been extensively investigated in the past. We show that a digraph with minimum outdegree a*n contains a directed triangle for a = 0.3465. The proof builds on arguments used to establish previously known bounds, due to Shen from 1998 (a = 0.3542) and Hamburger, Haxell and Kostochka from 2008 (a = 0.3531). It consists of a combination of ~80 computer generated inequalities. Based on joint work with Jan Hladky and Daniel Kral.

Series: Graph Theory Seminar

Tiling problems belong to the oldest problems in whole mathematics. They attracted attention of many famous mathematicians. Even one of the Hilbert problems is devoted to the topic. The interest in tilings by unit cubes originated with a conjecture raised by Minkowski in 1908. In this lecture we will discuss the conjecture, and other closely related problems.

Series: Graph Theory Seminar

Each graph can be embedded in 3-space. The problem becomes more interesting if we put restrictions on the type of embedding. For example, a linkless embedding of a graph is one where each pair of vertex-disjoint circuits has linking number equal to zero. The class of all graphs that have a linkless embedding is closed under taking minors. Robertson, Seymour, and Thomas gave the forbidden minors for this class of graphs. Open remained how to find a linkless embedding in polynomial time. In the talk we start with discussing an algorithm to find a linkless embedding.Instead of embedding the graph in 3-space, we could also consider mapping properties of certain superstructures of the graph in 3-space, and, indeed, if this superstructure has not the right mapping properties in 3-space, see whether it has the right one in 4-space, etc. Recently, we introduced for a graph G a new graph parameter \sigma(G), which is defined as the smallest d such that superstructures of G have a zero intersection mapping in d-space. The nicest property of this graph parameter is its independence of the superstructure and thus depends on the graph only. For d=2 and d=3, \sigma(G) \leq d if and only if G is outerplanar and planar, respectively. The graphs G with \sigma(G)\leq 4 are exactly those that have a linkless embedding. In the second part of the talk we will discuss this new graph parameter. (This part is joint work with R. Pendavingh.)

Series: Graph Theory Seminar

Given a configuration of pebbles on the vertices of a connected graph G, a pebbling move is defined as the removal of two pebbles from some vertex, and the placement of one of these on an adjacent vertex. The pebbling number of a graph G is the smallest integer k such that given any configuration of k pebbles on G and any specified vertex v in V(G), there is a sequence of pebbling moves that sends a pebble to v. We will show that the pebbling number of a graph of diameter four on n vertices is at most 3n/2 + O(1), and this bound is best possible up to an additive constant. This proof, based on a discharging argument and a decomposition of the graph into ''irreducible branches'', generalizes work of Bukh on graphs of diameter three. Further, we prove that the pebbling number of a graph on n vertices with diameter d is at most (2^{d/2} - 1)n + O(1). This also improves a bound of Bukh.