1,662 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
Dephasing by time-dependent random potentials
Diffusion of electrons in a two-dimensional system with time-dependent random
potentials is investigated numerically. The correction to the conductivity due
to inelastic scatterings by oscillating potentials is shown to be a universal
function of the frequency , which is consistent with the weak
localization prediction .Comment: 4 pages, RevTeX, 3 postscript figures, accepted in J. Phys. Soc.
Jpn.(1997)Ap
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