Some recent results on edge-colored graphs Shinya Fujita (Yokohama - PowerPoint PPT Presentation

Some recent results on edge-colored graphs Shinya Fujita (Yokohama City University) ☆ The topic is based on the following joint papers with my Chinese colleagues. ⚫ “Color degree and monochromatic degree conditions for short properly colored cycles in edge-colored graphs “ JGT 2018 (with Ruonan Li and Shinggui Zhang) ⚫ “On sufficient conditions for rainbow cycles in edge-colored graphs” DM, accepted (with Bo Ning, Chuandong Xu and Shenggui Zhang) ⚫ ”Decomposing edge-colored graphs under color degree constraints” CPC, accepted (with Ruonan Li and Guanghui Wang)

Part I: Degree results ⚫ “Color degree and monochromatic degree conditions for short properly colored cycles in edge-colored graphs “ JGT 2018 (with Ruonan Li and Shinggui Zhang) ⚫ “On sufficient conditions for rainbow cycles in edge-colored graphs” DM, accepted (with Bo Ning, Chuandong Xu and Shenggui Zhang) Part II: Decomposition results ⚫ ”Decomposing edge-colored graphs under color degree constraints” CPC, accepted (with Ruonan Li and Guanghui Wang)

Part I: Degree results ⚫ “Color degree and monochromatic degree conditions for short properly colored cycles in edge-colored graphs “ JGT 2018 (with Ruonan Li and Shinggui Zhang) ⚫ “On sufficient conditions for rainbow cycles in edge-colored graphs” DM, accepted (with Bo Ning, Chuandong Xu and Shenggui Zhang)

In this talk, we consider degree condition for cycles in edge-colored graphs. Let { G) } Sc ( G ) new min : ) = 1 dccv : color degree of v; i.e., the number of colors adjacent to v in G. Ex. G 84 G) 8 ( c ) : 3 te 8461=2 a

In this talk, we consider degree condition for cycles in edge-colored graphs. Let { G) } Sclc new min ) := ) 1 dccv § colored properly C ! 4 s , color degree of v; i.e., the number of colors adjacent to v in G. Ex. G 3846 ) 8( c) : Note 8461=2 a

In this talk, we consider degree condition for cycles in edge-colored graphs. Let { G) } Sc ( G ) new min : ) = 1 dccv : color degree of v; i.e., the number of colors adjacent to v in G. Ex. G 84 G) 8 ( c ) : 3 te 8461=2 a

In this talk, we consider degree condition for cycles in edge-colored graphs. Let { G) } Sc ( G ) new min : ) = 1 dccv § rainbow ! I Triangle color degree of v; i.e., the number of colors adjacent to v in G. Ex. G 84 G) 8 ( o ) : Note > 8461=2 a

For a vertex v in an edge-colored graph G, let CN(v) be the set of colors assigned to edges incident to v. v } )={ red CN green ( v , Ex. G

☆ Some natural questions: What is the sharp degree conditions for the followings? Prop. 1: If G is an edge-colored graph of order with n f- ( 89 G) , then G contains a properly colored cycle. 3 h ) Prop. 2. If G is an edge-colored graph of order with n . 38 ( 89 G) , then G contains a rainbow cycle. h )

☆ Answer for Prop.1 Prop. 1: If G is an edge-colored graph of order with n fin , then G contains a properly colored cycle. 846 ) 3 ) I ( 2018 Th ) Li F JGT , Zhang and , Fln ) Prop . I of least D value be the Sit true Let is . . ! !&÷ =D Then I holds n + .

Construction of sharpness example: Gi : Doing this way we Can , Gi from Gitl Construct Gz : 846in ) that itl So = PC cycle Gitt and has no - . - Gi Gi Note =D 8461 , ) : G : , } ! § of , IYGDH =D 62 Gz g

☆ Partial answer for Prop. 2 Prop. 2. If G is an edge-colored graph of order with n . 8 ( JCCG ) , then G contains a rainbow cycle. 3 h ) ( 2 EUJC ) Th Li 2014 al et . of . 2 value Prop Let D be The least fcn ) true is st . . zht holds 1 D Then < .

( EUJC ) 2 Th 2014 Li al et . Colored 75 of order edge graph Let G be N with an - 3 E 3 Then G= Ken 8. ( G) Go rainbow triangle , eh or . . ( 3 AUJC 2005 74 Th Broersma ) al et . Colored of order Let G edge graph ,t be n s an - , ICN U ) CN ( n ) for ;=Y Y V( G) ( 3 n 1 u C- 1 pair - every . C 4 3 rainbow ⇒ Triangle rainbow Go Then or .

Our results are following. F) ( Ning Th 4 and Xu Zhang , , Colored let of order For 1<31 6 be edge graph 7105k 24 n an - - , CN CN ( n ) U ) St ( 3 for ;=Y Y V( G) 1 1 n u pair C- 1 - every , . G C Then K rainbow D 4. F) 5 ( Ning Th and Xu Zhang , , Colored of order 6 Let 6 be graph edge St n 7 an - , CN U ) CN ( n ) ( 3 for ;=Y Y V( G) 1 n 1 u pair C- 1 - every . 3 GI Ken Then Go rainbow triangle , eh or .

Our results are following. ns.t F) ( Ning Th 6 and Zhang Xu , , Colored of order let 6 be For 1<31 edge graph an - , . ICN +641<+1 ( n ) I U ) CN HE ( 3 for ,±Y Y V( G) C- u pair every . cycles G) K disjoint rainbow Then vertex - . if G colored of N Cor edge graph order For 1<>-1 is an - . , 846 zh Then ) +64kt with 7 1 , G) cycles K disjoint rainbow vertex - .

Our results are following: ) ( and F JGT Th . 7 Li , Zhang 2018 ⇒ 84km If then Km n ) PC Cat Co 2 3 in or n , . , 2018 ) . 8 ( J and F GT Th Li , Zhang ⇒ 33 84km If then km PC Cat n ) in n . . , Remark. The minimum color degree conditions are sharp.

Our results are following: KPC 33k ) ( and F JGT Th . 7 Li , Zhang 2018 ⇒ 84km If km then n ) PC Cat Co 2 3 in or n , . , . , for 2018 ) . 8 ( J and F GT Th Li , Zhang ⇒ 84km If then Km n ) Cain n . . , Remark. The minimum color degree conditions are sharp.

I propose the following conjecture: Conj . 3 ¥n each 81km then If ,n ) vertex +1 colored of contained properly cycles is in , respectively 2h } { length 4.6 min ZM . . . . , , ,

We have the following partial result to this conjecture. . 9 ( ) Th F GT , Zhang and J 2018 Li Fit each then It 84km , n ) I vertex 3 colored cycle of contained properly is in a length 4 .

The bound on the color degree condition is best possible. . Coloring # edge Prop 81km of Km ,n ) ns.t . , . +3 Fue and Km properly St mtyn any = ,n : : Colored does contain C not 4 The case where m=5, n=4: 5+4+3 841<5,4 ) = 4

Part II: Decomposition results ⚫ ”Decomposing edge-colored graphs under color degree constraints” CPC, accepted (with Ruonan Li and Guanghui Wang)

I propose the following conjecture: Conj. 846 ) Let G be an edge-colored graph with atbtl 7 . Then G can be partitioned into 2 parts A and B s.t. 89643 ] ) 8961 A ] ) > b and > a . G B A ' 85,1 , 8 sizatbtt > a

B A Our main results are following. 832 85,2 G - Conj. is true for a=b=2. Thm. (Ruonan Li, Guanghui Wang, and F) 89 G) Let G be an edge-colored graph with 35 . Then G can be partitioned into 2 parts A and B s.t. 846 [ B ] ) 8 '(G[ A ] ) and 32 2 > , .

Our results are closely related to Bermond-Thomassen's conjecture in digraphs. Pbm. Determine the least value f(k) which makes the following proposition true. ft ( D ) f ( k ) Prop. Every digraph D with 3 contains k vertex-disjoint dicycles. Conj. (Bermond and Thomassen, JGT'81) f ( k ) 2k = 1 - . a. *• a. Known results: True for k ≦ 3. a. *• a. *•

In fact, we obtained a stronger statement. To state this, let g(k) be the following function. Glk )=( 11<=1 ) 2 ma×{ . 1) +3 } flk ) 1 k > 2) -11,91k Ref. Pbm. Determine the least value f(k) which makes the following proposition true. gt( D) 3f( k ) Prop. Every digraph D with contains k vertex-disjoint dicycles.

We obtained the following theorem. Thm 1. (Ruonan Li, Guanghui Wang and F.) 846 ) 8 ( k ) Let G be an edge-colored graph with > . Then G can be partitioned into k parts A1,...,Ak s.t. 8(G[ Ai ] ) 32 for K K if . G Note: g(2) = 5. 2 85.2 832 832

Proof idea for Theorem 1. In view of induction on k, we can check that proving the case k=2 is essential. Thm. (Ruonan Li, Guanghui Wang, and F) 89 G) 35 Let G be an edge-colored graph with . Then G can be partitioned into 2 parts A and B s.t. 846 [ B ] ) 8 '(G[ A ] ) and 32 2 > , .

It suffices to show that the following proposition is true. ' 8 75 Prop.1. If G is an edge-colored graph with (G) , then G has two vertex-disjoint subgraphs A1,A2 s.t. STAZ ) and 84A , ) 32 2 z . - Prop.1 implies our theorem. ° :) Take A and Azl Az Aiu that I maximum is so , . ' ( G- ( Aiu 8 G- 1 Aiu Suppose If Az ) Az ) ) then 32 ¥0 . , [ Ai G- A , ] desired 846 is But - I Aiu Az ) ) partition E 1 a , . contradict maximal of Azl The would Aiu ity 1 O .

- Prop.1 implies our theorem. °o° ) Take A and Azl Az Aiu that maximum is 1 so , . ' ( G- ( Aiu 8 G- Suppose lAiuAz If Az ) ) ) then ¥0 32 . , [ Ai G- A , ] desired is But 846 lAiuAz ) ) partition E 1 a , - . contradict maximal of IAIUAZI The would ity O . 2 2 z × × no two G- ( Aiuth )

Proof ideas: By contradiction, let G be a counterexample of Prop.1'. We choose such an edge-colored G so that: (i) |G| is as small as possible, and subject to (i); (ii) |E(G)| is as small as possible, and subject to (ii); (iii) the number of colors in G is as large as possible.