Girth-six conjecture for regular graphs

Establish whether every d-regular graph with d≥3 and girth at least 6 realizes the sequence (2^{d+1}).

Background

The paper proves that every d-regular graph of girth at least 7, for d≥3, realizes (2{d+1}), meaning that each of the d+1 color classes contains at least two color-dominating vertices. It also proves the same conclusion for cubic graphs of girth at least 6. These results motivate the conjecture that the girth-six conclusion holds for all degrees d≥3.

The conjecture would extend the cubic result to arbitrary regular degree and would show that the obstruction supplied by the 7-cycle is confined to degree 2. It remains unresolved in the paper.

References

This leads to the following conjecture. Let $d\geq 3$ and let $G$ be a $d$-regular graph of girth $g(G)\ge 6$. Then $G$ realizes the sequence $(2{d+1})$.

— Sequence b-colorings in graphs  (2609.08484 - Jakovac et al., 8 Sep 2026) in Conjecture 1, Section 4 (Regular graphs with prescribed girth)

Let $d\ge 2$, and let $S$ be a sequence of length $d+1$. What is the minimum integer $g$ such that every $d$-regular graph of girth at least $g$ realizes $S$?

— Sequence b-colorings in graphs  (2609.08484 - Jakovac et al., 8 Sep 2026) in Question 2, Section 7 (Concluding remarks and open questions)