tight product and semi-coloring of graphs
DESCRIPTION
Tight product and semi-coloring of graphs. Masaru Kamada Tokyo University of Science Graph Theory Conference i n honor of Yoshimi Egawa on the occasion his 60 th birthday September 10-14, 2013. In this talk, all graphs are finite, undirected and allowed multiple edges without loops. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/1.jpg)
Tight product and semi-coloring of graphs
Masaru KamadaTokyo University of Science
Graph Theory Conferencein honor of Yoshimi Egawa on the occasion his 60th birthday
September 10-14, 2013
![Page 2: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/2.jpg)
• In this talk, all graphs are finite, undirected and allowed multiple edges without loops.
No loopsMultiple edges
![Page 3: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/3.jpg)
![Page 4: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/4.jpg)
Example
{1}{2 }{3 }{1,2}{1,3 }{2,3 }
![Page 5: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/5.jpg)
The main result
![Page 6: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/6.jpg)
Almost regular graph
![Page 7: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/7.jpg)
![Page 8: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/8.jpg)
Outline of proof of Lemma 2
![Page 9: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/9.jpg)
![Page 10: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/10.jpg)
![Page 11: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/11.jpg)
![Page 12: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/12.jpg)
Outline of proof of Lemma 3
![Page 13: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/13.jpg)
![Page 14: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/14.jpg)
![Page 15: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/15.jpg)
![Page 16: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/16.jpg)
![Page 17: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/17.jpg)
Example of case II
![Page 18: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/18.jpg)
Example of subcase II-ii
![Page 19: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/19.jpg)
![Page 20: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/20.jpg)
{1}{2 }{3 }{1,2}{2,3 }
![Page 21: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/21.jpg)
{1}{2 }{3 }{1,2}{2,3 }
![Page 22: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/22.jpg)
{1}{2 }{3 }{1,2}{2,3 }
![Page 23: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/23.jpg)
![Page 24: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/24.jpg)
Tight product
![Page 25: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/25.jpg)
Example
25
![Page 26: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/26.jpg)
Example
26
![Page 27: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/27.jpg)
The existence of the tight product (1)
![Page 28: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/28.jpg)
![Page 29: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/29.jpg)
The existence of the tight product (2)
![Page 30: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/30.jpg)
![Page 31: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/31.jpg)
Thank you for your attention
![Page 32: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/32.jpg)
Proper-edge-coloring
![Page 33: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/33.jpg)
Classification of simple graphs
![Page 34: Tight product and semi-coloring of graphs](https://reader036.vdocument.in/reader036/viewer/2022062811/56816222550346895dd24c32/html5/thumbnails/34.jpg)
Example
Class-1 Class-2
1
1
32
3
2
15
4
5 3 3 1
2
42