Conjecture 189 on traceability under a bound on even-distance sets

Prove that every connected finite simple graph G satisfying max{|Even_G(v)| : v ∈ V(G)} ≤ d₂ + 1 is traceable, where Even_G(v) is the set of vertices at even distance from v and d₂ is the second-smallest vertex degree of G.

Background

The paper studies traceability, meaning the existence of a Hamiltonian path, in connected graphs subject to a restriction on the number of vertices at even distance from each vertex. For a graph G, Even_G(v) denotes the vertices whose distance from v is even, and d₂ denotes the second-smallest degree in the degree sequence of G.

The conjecture is identified as Conjecture 189 from Written on the Wall II. The paper establishes the claimed traceability under several additional hypotheses, including bounds on the diameter or order, a specified connectivity condition, and the absence of 2-connectivity. These results constitute partial progress, so the general conjecture remains unresolved in the paper.

References

The following is Conjecture 189 from Written on the Wall II (), and in this paper we make partial progress towards its resolution. Let $G=(V,E)$ be a connected graph such that \max{(v):v\in V}\le d_2+1. Then $G$ is traceable.

— Conditions for traceability under a bound on the size of even-distance sets  (2609.11719 - Easwar et al., 10 Sep 2026) in Section 1, Introduction and Preliminaries, immediately before Theorem 1 (Conjecture 189)