1,055 research outputs found
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)
Polynomial treewidth forces a large grid-like-minor
Robertson and Seymour proved that every graph with sufficiently large
treewidth contains a large grid minor. However, the best known bound on the
treewidth that forces an grid minor is exponential in .
It is unknown whether polynomial treewidth suffices. We prove a result in this
direction. A \emph{grid-like-minor of order} in a graph is a set of
paths in whose intersection graph is bipartite and contains a
-minor. For example, the rows and columns of the
grid are a grid-like-minor of order . We prove that polynomial
treewidth forces a large grid-like-minor. In particular, every graph with
treewidth at least has a grid-like-minor of order
. As an application of this result, we prove that the cartesian product
contains a -minor whenever has treewidth at least
.Comment: v2: The bound in the main result has been improved by using the
Lovasz Local Lemma. v3: minor improvements, v4: final section rewritte
A Variant of the Erd\H{o}s-S\'os Conjecture
A well-known conjecture of Erd\H{o}s and S\'os states that every graph with
average degree exceeding contains every tree with edges as a
subgraph. We propose a variant of this conjecture, which states that every
graph of maximum degree exceeding and minimum degree at least contains every tree with edges.
As evidence for our conjecture we show (i) for every there is a
such that the weakening of the conjecture obtained by replacing by
holds, and (ii) there is a such that the weakening of the conjecture
obtained by replacing by holds
A linear-time algorithm for finding a complete graph minor in a dense graph
Let g(t) be the minimum number such that every graph G with average degree
d(G) \geq g(t) contains a K_{t}-minor. Such a function is known to exist, as
originally shown by Mader. Kostochka and Thomason independently proved that
g(t) \in \Theta(t*sqrt{log t}). This article shows that for all fixed \epsilon
> 0 and fixed sufficiently large t \geq t(\epsilon), if d(G) \geq
(2+\epsilon)g(t) then we can find this K_{t}-minor in linear time. This
improves a previous result by Reed and Wood who gave a linear-time algorithm
when d(G) \geq 2^{t-2}.Comment: 6 pages, 0 figures; Clarification added in several places, no change
to arguments or result
Fast separation in a graph with an excluded minor
Let be an -vertex -edge graph with weighted vertices. A pair of vertex sets is a of if , there is no edge between and , and both and have weight at most the total weight of . Let be fixed. Alon, Seymour and Thomas [ 1990] presented an algorithm that in time, either outputs a -minor of , or a separation of of order . Whether there is a time algorithm for this theorem was left as open problem. In this paper, we obtain a time algorithm at the expense of separator. Moreover, our algorithm exhibits a tradeoff between running time and the order of the separator. In particular, for any given , our algorithm either outputs a -minor of , or a separation of with order in time
An upper bound for the chromatic number of line graphs
It was conjectured by Reed [reed98conjecture] that for any graph , the graph's chromatic number is bounded above by , where and are the maximum degree and clique number of , respectively. In this paper we prove that this bound holds if is the line graph of a multigraph. The proof yields a polynomial time algorithm that takes a line graph and produces a colouring that achieves our bound
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