On a tree-partition problem

Abstract

If T=(V,E) is a tree then – T denotes the additive hereditary property consisting of all graphs that does not contain T as a subgraph. For an arbitrary vertex v of T we deal with a partition of T into two trees T1, T2, so that V(T1)∩V(T2)={v}, V(T1)∪(T2)=V(T), E(T1)∩E(T2)=∅, E(T1)∪E(T2)=E(T), T[V(T1)\{v}]⊆E(T1) and T[V(T2)\{v}]⊆E(T2). We call such a partition a Tv−partition of T. We study the following em: Given a graph G belonging to –T. Is it true that for any Tv-partition T1, T2 of T there exists a partition {V1,V2} of the vertices of G such that G[V1]∈−T1 and G[V2]∈−T2? This problem provides a natural generalization of Δ-partition problem studied by L. Lovász ([L. Lovász, On decomposition of graphs. Studia Sci. Math. Hungar. 1 (1966) 237–238]) and Path Partition Conjecture formulated by P. Mihók ([P. Mihók, Problem 4, in: M. Borowiecki, Z. Skupien (Eds.), Graphs, Hypergraphs and Matroids, Zielona Góra, 1985, p. 86]). We present some partial results and a contribution to the Path Kernel Conjecture that was formulated with connection to Path Partition Conjecture.