249 research outputs found
Extremal Infinite Graph Theory
We survey various aspects of infinite extremal graph theory and prove several
new results. The lead role play the parameters connectivity and degree. This
includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure
On First-Order Definable Colorings
We address the problem of characterizing -coloring problems that are
first-order definable on a fixed class of relational structures. In this
context, we give several characterizations of a homomorphism dualities arising
in a class of structure
A note on circular chromatic number of graphs with large girth and similar problems
In this short note, we extend the result of Galluccio, Goddyn, and Hell,
which states that graphs of large girth excluding a minor are nearly bipartite.
We also prove a similar result for the oriented chromatic number, from which
follows in particular that graphs of large girth excluding a minor have
oriented chromatic number at most , and for the th chromatic number
, from which follows in particular that graphs of large girth
excluding a minor have
Logarithmically-small Minors and Topological Minors
Mader proved that for every integer there is a smallest real number
such that any graph with average degree at least must contain a
-minor. Fiorini, Joret, Theis and Wood conjectured that any graph with
vertices and average degree at least must contain a -minor
consisting of at most vertices. Shapira and Sudakov
subsequently proved that such a graph contains a -minor consisting of at
most vertices. Here we build on their method
using graph expansion to remove the factor and prove the
conjecture.
Mader also proved that for every integer there is a smallest real number
such that any graph with average degree larger than must contain
a -topological minor. We prove that, for sufficiently large , graphs
with average degree at least contain a -topological
minor consisting of at most vertices. Finally, we show
that, for sufficiently large , graphs with average degree at least
contain either a -minor consisting of at most
vertices or a -topological minor consisting of at most
vertices.Comment: 19 page
Small Complete Minors Above the Extremal Edge Density
A fundamental result of Mader from 1972 asserts that a graph of high average
degree contains a highly connected subgraph with roughly the same average
degree. We prove a lemma showing that one can strengthen Mader's result by
replacing the notion of high connectivity by the notion of vertex expansion.
Another well known result in graph theory states that for every integer t
there is a smallest real c(t) so that every n-vertex graph with c(t)n edges
contains a K_t-minor. Fiorini, Joret, Theis and Wood conjectured that if an
n-vertex graph G has (c(t)+\epsilon)n edges then G contains a K_t-minor of
order at most C(\epsilon)log n. We use our extension of Mader's theorem to
prove that such a graph G must contain a K_t-minor of order at most
C(\epsilon)log n loglog n. Known constructions of graphs with high girth show
that this result is tight up to the loglog n factor
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