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On the Weisfeiler-Leman dimension of some polyhedral graphs
Authors
Haiyan Li
Ilia Ponomarenko
Peter Zeman
Publication date
26 May 2023
Publisher
View
on
arXiv
Abstract
Let
m
m
m
be a positive integer,
X
X
X
a graph with vertex set
Ω
\Omega
Ω
, and
W
L
m
(
X
)
{\rm WL}_m(X)
WL
m
​
(
X
)
the coloring of the Cartesian
m
m
m
-power
Ω
m
\Omega^m
Ω
m
, obtained by the
m
m
m
-dimensional Weisfeiler-Leman algorithm. The
W
L
{\rm WL}
WL
-dimension of the graph
X
X
X
is defined to be the smallest
m
m
m
for which the coloring
W
L
m
(
X
)
{\rm WL}_m(X)
WL
m
​
(
X
)
determines
X
X
X
up to isomorphism. It is known that the
W
L
{\rm WL}
WL
-dimension of any planar graph is
2
2
2
or
3
3
3
, but no planar graph of
W
L
{\rm WL}
WL
-dimension
3
3
3
is known. We prove that the
W
L
{\rm WL}
WL
-dimension of a polyhedral (i.e.,
3
3
3
-connected planar) graph
X
X
X
is at most
2
2
2
if the color classes of the coloring
W
L
2
(
X
)
{\rm WL}_2(X)
WL
2
​
(
X
)
are the orbits of the componentwise action of the group
A
u
t
(
X
)
{\rm Aut}(X)
Aut
(
X
)
on
Ω
2
\Omega^2
Ω
2
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oai:arXiv.org:2305.17302
Last time updated on 02/06/2023