circular coloring, orientations, critical cycles, and weighted digraphs
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Circular coloring, orientations, critical cycles, and weighted digraphs. Hong-Gwa Yeh ( 葉鴻國 ) Department of Mathematics National Central University [email protected]. 2008/01/25. k -coloring of a graph G. k -coloring of a graph G. (k, 1 )-coloring of a graph G. - PowerPoint PPT PresentationTRANSCRIPT
Circular coloring, orientations, critical cycles, and weighted digraphs
Hong-Gwa Yeh (葉鴻國 )Department of Mathematics National Central [email protected]
2008/01/25
k-coloring of a graph G
k-coloring of a graph G
(k,1)-coloring of a graph G
11
1
1
Zk
(k,d)-coloring of a graph G
Circular r-coloring of a graph G
Sr is a cycle with perimeter r
χ(G) and χc(G)
(k,1)-coloring}
}
, C C
C
●
●
●
●
●
●
●
| | | |max ,
| | |
7( , )
3|
C C
C CC
Minty’s Theorem
Theorem
is a cycle of G
an acyclic orientation of s.t.
( , )maxC
G
C k
has a -coloringG k
| | | |max ,
| |(
|, )
|
C C
CC
C
Minty’s Theorem
Theorem
an acyclic orientation of s.t.
| |
| |maxC
G
Ck
C
has a -coloringG k
is a cycle of G
an acyclic orientation of s.t.
( , )maxC
G
C k
( , )-coloring has a k dG has a ( ,1)-coloringG k
Minty’s Theorem
Theorem
an acyclic orientation of s.t.
| |
| |maxC
G
Ck
C
an acyclic orientation of s
|
|
.t.
|
|maxC
G
C k
C d
Generalized Minty’s Theorem
Minty’s Theorem
Generalized Minty’s Theorem
Revisit Minty’s Theorem
Theorem
an acyclic orientation of s.t.
| |
| |maxC
G
Ck
C
has a -coloringG k
Note: The max above is taken over all simple cycles of G.
Question: Could we reduce the number of cycles that need to be checked?
Tuza’s Theorem JCT(B), 1992
Theorem
an acyclic orientation of s.t.
| |
| |maxC
G
Ck
C
has a -coloringG k
Minty’s Theorem:
| 1(mod )|
| |
| |
an acyclic orientation of s.t.
maxC k
G
Ck
C
Revisit Generalized Minty’s Theorem
Theorem
an acyclic orientation of s
|
|
.t.
|
|maxC
G
d
C k
C
has a ( , )-coloringG k d
Note: The max above is taken over all simple cycles of G.
Question: Could we reduce the number of cycles that need to be checked?
Theorem:Generalized Minty’s Theorem:
Zhu’s Theorem JCT(B), 2002
an acyclic orientation of s
|
|
.t.
|
|maxC
G
d
C k
C
has a ( , )-coloringG k d
1 | |( mod ) 2 1
an acyclic orientation of s.t
| |
| |
.
maxd C k d
C
C
G
k
d
What comes next ?
circular r-coloring & (k,d)-coloring
circular r-coloring & (k,d)-coloring
Zhu’s Theorem: has a ( , )-coloringG k d
1 | |( mod ) 2 1
an acyclic orientation of s.t
| |
| |
.
maxd C k d
C
C
G
k
d
Folklore Theorem: circular - has colo n a ri grG
an acyclic orientation of s.t.
| |
| |maxC
G
Cr
C
?
Circular p-coloring of a digraph
Circular p-coloring of an edge-weighted digraph
Mohar (JGT, 2003)
Mohar’s Theorem (JGT,
2003)
Theorem:Let ( , ) be an edge-wei symmetricghted digr aphG c
,,,,,,,,,,,,,,
( , ) has a circular -coloringG c r--------------
a mapping ha: {0 ving,1} T E --------------
for each arc 1 xy yx xT T y --------------
for each dicycle | o>0 f |T C GC--------------
| |
| |maxC
c
T
C
Crs.t.
Question: Could we reduce the number of dicycles that need to be checked?
Our result 2007
Theorem:
( , ) has a circular -coloringG c r--------------
a mapping ha: {0 ving,1} T E --------------
for each arc 1 xy yx xT T y --------------
for each dicycle
with 0 |
| | >0
| (mod )T
c
C
C L
C
r
0 | | (mod )
| |
| |maxcC r L
c
T
Cr
C
s.t. is an arc of G
{ }max xy yxxy
c cL --------------
where
0 | | (mod ) 2C r
circular r-coloring & (k,d)-coloring
Zhu’s Theorem: has a ( , )-coloringG k d
1 | |( mod ) 2 1
an acyclic orientation of s.t
| |
| |
.
maxd C k d
C
C
G
k
d
Folklore Theorem: circular - has colo n a ri grG
an acyclic orientation of s.t.
| |
| |maxC
G
Cr
C
?
Corollary:
Zhu’s Theorem:
Proof:
Sorry, I am running out of time. You will see the definition in the problem section.
So, what is the definition of critical cycle appeared in your title ?
Thank you for your attention