61 research outputs found
Coloring Graphs with Forbidden Minors
Hadwiger's conjecture from 1943 states that for every integer , every
graph either can be -colored or has a subgraph that can be contracted to the
complete graph on vertices. As pointed out by Paul Seymour in his recent
survey on Hadwiger's conjecture, proving that graphs with no minor are
-colorable is the first case of Hadwiger's conjecture that is still open. It
is not known yet whether graphs with no minor are -colorable. Using a
Kempe-chain argument along with the fact that an induced path on three vertices
is dominating in a graph with independence number two, we first give a very
short and computer-free proof of a recent result of Albar and Gon\c{c}alves and
generalize it to the next step by showing that every graph with no minor
is -colorable, where . We then prove that graphs with no
minor are -colorable and graphs with no minor are
-colorable. Finally we prove that if Mader's bound for the extremal function
for minors is true, then every graph with no minor is
-colorable for all . This implies our first result. We believe
that the Kempe-chain method we have developed in this paper is of independent
interest
A Note on Hadwiger's Conjecture
Hadwiger's Conjecture states that every -minor-free graph is
-colourable. It is widely considered to be one of the most important
conjectures in graph theory. If every -minor-free graph has minimum
degree at most , then every -minor-free graph is
-colourable by a minimum-degree-greedy algorithm. The purpose of
this note is to prove a slightly better upper bound
Disproof of the List Hadwiger Conjecture
The List Hadwiger Conjecture asserts that every -minor-free graph is
-choosable. We disprove this conjecture by constructing a
-minor-free graph that is not -choosable for every integer
Forcing a sparse minor
This paper addresses the following question for a given graph : what is
the minimum number such that every graph with average degree at least
contains as a minor? Due to connections with Hadwiger's Conjecture,
this question has been studied in depth when is a complete graph. Kostochka
and Thomason independently proved that . More generally,
Myers and Thomason determined when has a super-linear number of
edges. We focus on the case when has a linear number of edges. Our main
result, which complements the result of Myers and Thomason, states that if
has vertices and average degree at least some absolute constant, then
. Furthermore, motivated by the case when
has small average degree, we prove that if has vertices and edges,
then (where the coefficient of 1 in the term is best
possible)
Densities of Minor-Closed Graph Families
We define the limiting density of a minor-closed family of simple graphs F to
be the smallest number k such that every n-vertex graph in F has at most
kn(1+o(1)) edges, and we investigate the set of numbers that can be limiting
densities. This set of numbers is countable, well-ordered, and closed; its
order type is at least {\omega}^{\omega}. It is the closure of the set of
densities of density-minimal graphs, graphs for which no minor has a greater
ratio of edges to vertices. By analyzing density-minimal graphs of low
densities, we find all limiting densities up to the first two cluster points of
the set of limiting densities, 1 and 3/2. For multigraphs, the only possible
limiting densities are the integers and the superparticular ratios i/(i+1).Comment: 19 pages, 4 figure
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