Path kernels and partitions

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

2. Coloring G=(V,E) (G)=k+l V1,V2: V1V2= V1V2= V(G[V1])≤k (G[V2])≤l V1 V2 k=2l=2

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

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

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

6. Difference between kernel and semikernel x4 x6 x7 x2 x1 x3 x8 x5 M={x2,x3,x4,x5} -isMP5-kernel in G? No Yes -isMP5-semikernel in G?

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

8. S Conjecture about semikernels G=(V,E) For every n≥2existsSV, thatSis Pn-semikernel in G. n=4

9. Conjectures: Every graphis -partitionable Every graph has Pn-kernelfor every n≥2 Every graph has Pn-semikernel for everyn≥2

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

11. The existence of Pn-kernels for small values of n Dunbar,Frick: Every graph has P7-kernel Meľnikov,Petrenko: Every graph has P8-kernel P.K.(2005): Every graph hasP9-kernel

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

13. Cycle lengths and path kernels • If G is graph with g(G)≥n-2, then G has Pn-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. Absent of Pn-kernels for big values of n Thomassen: Exists graph, that don’t have a P364-kernel P.K.(2005): For every n≥364exists graph, that don’t have a Pn-kernel

15. Algorithm to find Pn-semikernel for small values of n Let S=Hi, where i is smallest integer such that Hi is a subgraph of G. Algorithm: Initially we let B=V(G)-S and A=. • Identify all P8 terminal vertices of S and move all their B-neighbours to A. If N(S) ∩ B is empty, then stop, else 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. • If some P7-terminal vertex x of S has a B-neighbour, then move one B-neighbour of x to S and return to 1, else 4. • … 5 … 6… • If some P3-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. Sequence of graphs Hifor n=7 H1 H2 H3 H4 H5 Príklad H6 H7 H8 H9

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

18. Thanks for your attention.