MPII20041006
On the Hadwiger's conjecture for graph products
Chandran, L. Sunil and Sivadasan, Naveen
2004, 12 pages.
.
Status: available  back from printing
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),
where
chi(G) is the chromatic number of G. It is wellknown
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{d1}{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 wellknown graph products like ddimensional hypercubes,
Hamming graphs and the ddimensional grids. In particular, we show that
for a ddimensional hypercube Hd,
$2^{\lfloor\frac{d1}{2}\rfloor} <= m(Hd) <= 2^{\frac{d}{2}}\sqrt{d} +1$.
We also derive similar bounds for G^d for almost all
G with n nodes and at least nlog(n) edges.

 Attachement: MPII20041006.ps (137 KBytes)
URL to this document: https://domino.mpiinf.mpg.de/internet/reports.nsf/NumberView/20041006
BibTeX
@TECHREPORT{ChandranSivadasan2004,
AUTHOR = {Chandran, L. Sunil and Sivadasan, Naveen},
TITLE = {On the Hadwiger's conjecture for graph products},
TYPE = {Research Report},
INSTITUTION = {MaxPlanckInstitut f{\"u}r Informatik},
ADDRESS = {Stuhlsatzenhausweg 85, 66123 Saarbr{\"u}cken, Germany},
NUMBER = {MPII20041006},
YEAR = {2004},
ISSN = {0946011X},
}