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On the Hadwiger's conjecture for graph products

Chandran, L. Sunil and Sivadasan, Naveen

MPI-I-2004-1-006. 2004, 12 pages. | Status: available - back from printing | Next --> Entry | Previous <-- Entry

Abstract in LaTeX format:
The Hadwiger number m(G) of a graph G is the
largest integer h such that the complete graph on h nodes is a minor
of G. Equivalently, it is the largest integer such
that any graph on at most m(G) nodes is a minor of G.
The Hadwiger's conjecture states that for any graph G, m(G) >= chi(G),
chi(G) is the chromatic number of G. It is well-known
that for any connected undirected graph G, there exists a unique prime
factorization with respect to Cartesian graph products.
If the unique prime factorization of G is given as
G1 X G2 X ... X Gk, where each Gi is prime,
then we say that the product dimension of G is k.
Such a factorization can be computed efficiently.

In this paper, we study the Hadwiger's conjecture for graphs in terms
of their prime factorization.
We show that the Hadwiger's conjecture is true for a graph G if
the product dimension of G is at least 2log(chi(G)) + 3.
In fact, it is enough for G to have a connected graph M as a minor
whose product dimension is at
least 2log(chi(G)) + 3, for G to satisfy the Hadwiger's conjecture.
We show also that
if a graph G is isomorphic to F^d for some F, then
mr(G) >= chi(G)^{\lfloor \frac{d-1}{2} \rfloor}, and thus
G satisfies the Hadwiger's conjecture when d >= 3.
For sufficiently large d, our lower bound is exponentially higher
than what is implied by the Hadwiger's conjecture.

Our approach also yields (almost) sharp lower bounds for the
Hadwiger number of well-known graph products like d--dimensional hypercubes,
Hamming graphs and the d--dimensional grids. In particular, we show that
for a d--dimensional hypercube Hd,
$2^{\lfloor\frac{d-1}{2}\rfloor} <= m(Hd) <= 2^{\frac{d}{2}}\sqrt{d} +1$.
We also derive similar bounds for G^d for almost all
G with n nodes and at least nlog(n) edges.
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  AUTHOR = {Chandran, L. Sunil and Sivadasan, Naveen},
  TITLE = {On the Hadwiger's conjecture for graph products},
  TYPE = {Research Report},
  INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},
  ADDRESS = {Stuhlsatzenhausweg 85, 66123 Saarbr{\"u}cken, Germany},
  NUMBER = {MPI-I-2004-1-006},
  YEAR = {2004},
  ISSN = {0946-011X},