Circular coloring, orientations, critical cycles, and weighted digraphs

1. Circular coloring, orientations, critical cycles, and weighted digraphs Hong-Gwa Yeh (葉鴻國) Department of Mathematics National Central University hgyeh@math.ncu.edu.tw 2008/01/25

2. k-coloring of a graph G

3. k-coloring of a graph G

4. (k,1)-coloring of a graph G (k,d)-coloring of a graph G 1 1 1 Zk 1

5. Circular r-coloring of a graph G Sr is a cycle with perimeterr

6. (k,1)-coloring} } χ(G) and χc(G)

7. ● ● ● ● ● ● ●

8. Minty’s Theorem Theorem 

9. Minty’s Theorem Theorem 

10. Generalized Minty’s Theorem Minty’s Theorem Theorem 

11. Minty’s Theorem Generalized Minty’s Theorem

12. Revisit Minty’s Theorem Theorem  Note: The max above is taken over all simple cycles of G. Question: Could we reducethe number of cycles that need to be checked?

13. Tuza’s Theorem JCT(B), 1992 Minty’s Theorem: Theorem 

14. Revisit Generalized Minty’s Theorem Theorem  Note: The max above is taken over all simple cycles of G. Question: Could we reducethe number of cycles that need to be checked?

15. Zhu’s Theorem JCT(B), 2002 Generalized Minty’s Theorem: Theorem: 

16. What comes next ?

17. circular r-coloring & (k,d)-coloring

18. circular r-coloring & (k,d)-coloring Zhu’s Theorem:  Folklore Theorem:  ?

19. Circular p-coloring of a digraph

21. Circular p-coloring of an edge-weighted digraph Mohar (JGT, 2003)

22. Mohar’s Theorem(JGT, 2003) Theorem:  s.t. Question: Could we reducethe number of dicycles that need to be checked?

23. Our result 2007 Theorem:  s.t. where

24. circular r-coloring & (k,d)-coloring Zhu’s Theorem:  Folklore Theorem: Corollary:  ?

25. Proof: Zhu’s Theorem:

27. 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.

28. Thank you for your attention