circular coloring, orientations, critical cycles, and weighted digraphs

Circular coloring, orientations, critical cycles, and weighted digraphs

Hong-Gwa Yeh (葉鴻國 )Department of Mathematics National Central Universityhgyeh@math.ncu.edu.tw

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