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,d1)-coloring of a graph G
1
Zk
1
1
1
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
●
●
●
●  (C,  )  max  | C | , | C |   7
 
 
| C | | C |  3
Minty’s Theorem
Theorem G has a k -coloring
  an acyclic orientation 
max
of G s.t.
 (C ,  )  k
C is a cycle of G
 |C | |C | 
 (C ,  )  max   ,  
 | C | | C | 
Minty’s Theorem
Theorem G has a k -coloring
 an
an acyclic
acyclic orientation
orientation 

 
| C|(C ,  )  k
k
max
max

C isCa cycle|ofCG |

of
of G
G s.t.
s.t.
Generalized
Theorem
Minty’s Minty’s
Theorem
TheoremGGhas
hasa a(k(,1)-coloring
k , d )-coloring
.t.
  an acyclic orientation  of G ss.t.
|C | k
k
max

| C | d
C
Minty’s Theorem
Generalized
Minty’s Theorem
Revisit Minty’s Theorem
Theorem G has a k -coloring
  an
acyclic orientation  of G s.t.
|C |
k
max

| C |
C
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
Minty’s
Theorem G has a k -coloring
Theorem:
 an
anacyclic
acyclicorientation
orientation of
ofGGs.t.
s.t.
|C
| C| |
k
max
  k
| C| |
C k )| C
|C | 1(mod
Revisit
Generalized Minty’s Theorem
Theorem G has a (k , d )-coloring
  an acyclic orientation  of G s.t.
|C | k

max

| C | d
C
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?
Zhu’s Theorem JCT(B), 2002
Generalized
Theorem:
Minty’s
Theorem: G
has a (k , d )-coloring
acyclic orientation  of G s.t
s.t..
| C | |kC | k
  
max
max

|C | d
1 d |C |(mod|kC
)2
d | 1 d 
  an
What comes next ?
circular r-coloring & (k,d)-coloring
circular r-coloring & (k,d)-coloring
Zhu’s Theorem: G has a (k , d )-coloring
  an acyclic orientation  of G s.t.
max
1 d |C |(mod k )  2 d 1
|C | k


| C | d
Folklore Theorem: G has a circular r -coloring

 an acyclic orientation  of G s.t.
|C |
r
max

| C |
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 (G, c) be an edge-weighted symmetric digraph
(G,c) has a circular r -coloring

 a mapping T : E  {0,1} having
 Txy  Tyx  1 for each arc xy
 | C |T >0 for each dicycle C of G
| C |c
r
s.t.
max
| Cnumber
|T
Question: Could we reduce
the
of dicycles
C
that need to be checked?
Our result
2007
Theorem:
(G,c) has a circular r -coloring

 a mapping T : E  {0,1} having
 Txy  Tyx  1 for each arc xy
 | C |T >0 for each dicycle C
with 0 | C |c (mod r )  L
| C |c

r
max
s.t.
where L 0|C| (mod r ) L | C{c|Txy  c yx }
c
max
xy is an arc of G
circular r-coloring & (k,d)-coloring
Zhu’s Theorem: G has a (k , d )-coloring
  an acyclic orientation  of G s.t.
max
1 d |C |(mod k )  2 d 1
|C | k


| C | d
Folklore Corollary:
Theorem: G has a circular r -coloring

 an acyclic orientation  of G s.t.
|C |
r
max

| C |
C
?
0 | C | (mod r )  2
Zhu’s Theorem:
Proof:
So, what is the definition
of critical cycle appeared
in your title ?
Sorry, I am
running out of
time. You will see
the definition in
the problem
section.
Thank you for your attention