Path Partition Conjecture

Prove the Path Partition Conjecture: for every graph G with detour order \(\tau(G)=a+b\), construct a partition \(V_1,V_2\) of \(V(G)\) such that \(\tau(G\langle V_1\rangle)\leq a\) and \(\tau(G\langle V_2\rangle)\leq b\).

Background

The thesis introduces the Path Partition Conjecture as a conjecture relating the detour order of a graph to partitions of its vertex set. The conjecture is connected to detour chromatic numbers and is cited as a topic discussed in earlier work.

References

This conjecture states that, for any graph $G$, if $\tau(G) = a+b$ then there exists a partition $V_1$, $V_2$ of $V(G)$ such that $\tau(G\langle V_1 \rangle) \le a$ and $\tau(G\langle V_2 \rangle) \le b$.

— Detours In Graphs  (2507.12086 - Bullock, 16 Jul 2025) in Section 1, Section "Background"