On a cycle partition problem

Let G be any graph and let c(G) denote the circumference of G. We conjecture that for every pair c₁,c₂ of positive integers satisfying c₁+c₂=c(G), the vertex set of G admits a partition into two sets V₁ and V₂, such that V_i induces a graph of circumference at most c_i, i=1,2. We establish various results in support of the conjecture; e.g. it is observed that planar graphs, claw-free graphs, certain important classes of perfect graphs, and graphs without too many intersecting long cycles, satisfy the conjecture.This work is inspired by a well-known, long-standing, analogous conjecture involving paths.