179 research outputs found
Hadwiger Number and the Cartesian Product Of Graphs
The Hadwiger number mr(G) of a graph G is the largest integer n for which the
complete graph K_n on n vertices is a minor of G. Hadwiger conjectured that for
every graph G, mr(G) >= chi(G), where chi(G) is the chromatic number of G. In
this paper, we study the Hadwiger number of the Cartesian product G [] H of
graphs.
As the main result of this paper, we prove that mr(G_1 [] G_2) >= h\sqrt{l}(1
- o(1)) for any two graphs G_1 and G_2 with mr(G_1) = h and mr(G_2) = l. We
show that the above lower bound is asymptotically best possible. This
asymptotically settles a question of Z. Miller (1978).
As consequences of our main result, we show the following:
1. Let G be a connected graph. Let the (unique) prime factorization of G be
given by G_1 [] G_2 [] ... [] G_k. Then G satisfies Hadwiger's conjecture if k
>= 2.log(log(chi(G))) + c', where c' is a constant. This improves the
2.log(chi(G))+3 bound of Chandran and Sivadasan.
2. Let G_1 and G_2 be two graphs such that chi(G_1) >= chi(G_2) >=
c.log^{1.5}(chi(G_1)), where c is a constant. Then G_1 [] G_2 satisfies
Hadwiger's conjecture.
3. Hadwiger's conjecture is true for G^d (Cartesian product of G taken d
times) for every graph G and every d >= 2. This settles a question by Chandran
and Sivadasan (They had shown that the Hadiwger's conjecture is true for G^d if
d >= 3.)Comment: 10 pages, 2 figures, major update: lower and upper bound proofs have
been revised. The bounds are now asymptotically tigh
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
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