A note on Vizing's conjecture · arXivDesk2212.09571Dec 19, 2022
A note on Vizing's conjecture
Simon Špacapan
Abstract
Let γ(G) denote the domination number of graph G. Let G and H be graphs and G□H their Cartesian product. For h∈V(H)
define
Gh={(g,h)∣g∈V(G)} and call this set a
-layer of
. We prove the following special case of Vizing's conjecture. Let
be a dominating set of
. If there exist minimum dominating sets
and
of
such that for every
, the projection of
to
is contained in
or
, then
∣D∣≥γ(G)γ(H) .