1. Path kernels and partitions Peter Katrenič Institute of Mathematics Faculty of Science P. J. Šafárik University, Košice

2. 31.5.20052 Coloring G=(V,E) V1V1 V2V2  V 1,V 2 : V 1  V 2 =  V 1  V 2 = V  (G[V 1 ])≤k  (G[V 2 ])≤l  (G)=k+l k=2 l=2

3. 31.5.20053 Graph decomposition with bounded maximal degree G=(V,E) V1V1 V2V2  V1,V2: V1  V2=  V1  V2= V  (G[V1])≤k  (G[V2])≤l  (G)=k+l k=2 l=3

4. 31.5.20054 Path partition conjecture G=(V,E) V1V1 V2V2  V1,V2: V1  V2=  V1  V2= V  (G[V1])≤k  (G[V2])≤l  (G)=k+l k=3 l=5

5. 31.5.20055 P n -kernel and P n -semikernel A subset S of V(G) is called a P n -semikernel of G if: 1.  (G[S]) ≤ n-1 2. every vertex in N(S)-S is adjacent to a P n-1 -terminal vertex of G[S]. A subset K of V(G) is called P n -kernel of G if: 1.  (G[K]) ≤ n-1 2. every vertex v  V(G-K) is adjacent to a P n-1 -terminal vertex of G[K].

6. 31.5.20056 Difference between kernel and semikernel x1x1 x3x3 x2x2 x6x6 x7x7 x8x8 x5x5 x4x4 No Yes M={x 2,x 3,x 4,x 5 } -is M P 5 -kernel in G? -is M P 5 -semikernel in G?

7. 31.5.20057 K Conjecture about kernels G =(V,E) For every n≥2 exists K  V, that K is P n -kernel in G. n=4

8. 31.5.20058 S Conjecture about semikernels For every n ≥2 exists S  V, that S is P n -semikernel in G. G =(V,E) n=4

9. 31.5.20059 Conjecture s: Every graph is  -partitionable Every graph has P n -kernel for every n≥2 Every graph has P n -semikernel for every n≥2

10. 31.5.200510 Relationship among path kernels, semikernels and partitions Let P bea hereditary class of graphs. If every graph in P has P n -semikernel, then every graph in P has P n -kernel. Let G be a graph with  (G)=a+b, a≤b. If G has P b+1 -semikernel, then G is (a,b)-partitionable.

11. 31.5.200511 The existence of P n -kernels for small values of n Dunbar,Frick: Every graph has P 7 -kernel Meľnikov,Petrenko: Every graph has P 8 -kernel P.K.(2005): Every graph has P 9 -kernel

12. 31.5.200512 Graphs that have P n -kernels for all n If every block of graph G is either a complete graph or a cycle, then G has P n -kernel for all n≥2. Let G be a complete multipartite graph. The G has P n -kernel for all n≥2.

13. 31.5.200513 Cycle lengths and path kernels If G is graph with g(G)≥n-2, then G has P n -kernel Let G be a graph with  (G)=a+b, a≤b. If g(G)≥a-1, then G is (a,b)-partitionable Let G be a graph with  (G)=a+b, a≤b. If c(G)≤a+1, then G is (a,b)-partitionable

14. 31.5.200514 Absent of P n -kernels for big values of n Thomassen: Exists graph, that don’t have a P 364 -kernel P.K.(2005): For every n≥364 exists graph, that don’t have a P n -kernel

15. 31.5.200515 Algorithm to find P n -semikernel for small values of n Let S=H i, where i is smallest integer such that H i is a subgraph of G. Algorithm: Initially we let B=V(G)-S and A= . 1. Identify all P 8 terminal vertices of S and move all their B-neighbours to A. If N(S) ∩ B is empty, then stop, else 2. 2. If two vertices x and y in S have a common B-neighbour, then move one common B-neighbour of x and y to S and return to 1. 3. If some P 7 -terminal vertex x of S has a B-neighbour, then move one B- neighbour of x to S and return to 1, else 4. 4. … 5 … 6… 7. If some P 3 -terminal vertex x of S has a B-neighbour, then move one B- neighbour of x to S and return to 1, else 2.

16. 31.5.200516 Sequence of graphs H i for n=7 H1H1 H2H2 H3H3 H4H4 H5H5 H6H6 H7H7 H8H8 H9H9 Príklad

17. 31.5.200517 Summary of conjecture status Conjuncture about kernels is true for every n ≤9 a is false for every n ≥ 360 If G is graph with  (G)≤17, then G is  -partitionable

18. 31.5.200518 Thanks for your attention.