5,476 research outputs found
Graph removal lemmas
The graph removal lemma states that any graph on n vertices with o(n^{v(H)})
copies of a fixed graph H may be made H-free by removing o(n^2) edges. Despite
its innocent appearance, this lemma and its extensions have several important
consequences in number theory, discrete geometry, graph theory and computer
science. In this survey we discuss these lemmas, focusing in particular on
recent improvements to their quantitative aspects.Comment: 35 page
Topics in graph colouring and graph structures
This thesis investigates problems in a number of different areas of graph theory. These problems are related in the sense that they mostly concern the colouring or structure of the underlying graph. The first problem we consider is in Ramsey Theory, a branch of graph theory stemming from the eponymous theorem which, in its simplest form, states that any sufficiently large graph will contain a clique or anti-clique of a specified size. The problem of finding the minimum size of underlying graph which will guarantee such a clique or anti-clique is an interesting problem in its own right, which has received much interest over the last eighty years but which is notoriously intractable. We consider a generalisation of this problem. Rather than edges being present or not present in the underlying graph, each is assigned one of three possible colours and, rather than considering cliques, we consider cycles. Combining regularity and stability methods, we prove an exact result for a triple of long cycles. We then move on to consider removal lemmas. The classic Removal Lemma states that, for n sufficiently large, any graph on n vertices containing o(n^3) triangles can be made triangle-free by the removal of o(n^2) edges. Utilising a coloured hypergraph generalisation of this result, we prove removal lemmas for two classes of multinomials. Next, we consider a problem in fractional colouring. Since finding the chromatic number of a given graph can be viewed as an integer programming problem, it is natural to consider the solution to the corresponding linear programming problem. The solution to this LP-relaxation is called the fractional chromatic number. By a probabilistic method, we improve on the best previously known bound for the fractional chromatic number of a triangle-free graph with maximum degree at most three. Finally, we prove a weak version of Vizing's Theorem for hypergraphs. We prove that, if H is an intersecting 3-uniform hypergraph with maximum degree D and maximum multiplicity m, then H has at most 2D+m edges. Furthermore, we prove that the unique structure achieving this maximum is m copies of the Fano Plane
Extremal results in sparse pseudorandom graphs
Szemer\'edi's regularity lemma is a fundamental tool in extremal
combinatorics. However, the original version is only helpful in studying dense
graphs. In the 1990s, Kohayakawa and R\"odl proved an analogue of Szemer\'edi's
regularity lemma for sparse graphs as part of a general program toward
extending extremal results to sparse graphs. Many of the key applications of
Szemer\'edi's regularity lemma use an associated counting lemma. In order to
prove extensions of these results which also apply to sparse graphs, it
remained a well-known open problem to prove a counting lemma in sparse graphs.
The main advance of this paper lies in a new counting lemma, proved following
the functional approach of Gowers, which complements the sparse regularity
lemma of Kohayakawa and R\"odl, allowing us to count small graphs in regular
subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse
extensions of several well-known combinatorial theorems, including the removal
lemmas for graphs and groups, the Erd\H{o}s-Stone-Simonovits theorem and
Ramsey's theorem. These results extend and improve upon a substantial body of
previous work.Comment: 70 pages, accepted for publication in Adv. Mat
Pebbling in Semi-2-Trees
Graph pebbling is a network model for transporting discrete resources that
are consumed in transit. Deciding whether a given configuration on a particular
graph can reach a specified target is -complete, even for diameter
two graphs, and deciding whether the pebbling number has a prescribed upper
bound is -complete. Recently we proved that the pebbling number
of a split graph can be computed in polynomial time. This paper advances the
program of finding other polynomial classes, moving away from the large tree
width, small diameter case (such as split graphs) to small tree width, large
diameter, continuing an investigation on the important subfamily of chordal
graphs called -trees. In particular, we provide a formula, that can be
calculated in polynomial time, for the pebbling number of any semi-2-tree,
falling shy of the result for the full class of 2-trees.Comment: Revised numerous arguments for clarity and added technical lemmas to
support proof of main theorem bette
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