Multiplicative control of the equitable chromatic number by the lower bound

Determine whether there exists a constant C>0 such that every connected block graph G satisfies the multiplicative bound χ₌(G)≤C L(G), where χ₌(G) is the equitable chromatic number and L(G)=max{ω(G),(|V(G)|+1)/(αₘᵢₙ(G)+1)} with αₘᵢₙ(G)=min_{v∈V(G)}α(G,v).

Background

The paper disproves the previously conjectured additive bound χ₌(G)≤L(G)+1 for block graphs and proves the stronger result that χ₌(G)−L(G) is unbounded even for connected block graphs. Consequently, no absolute additive constant can control the equitable chromatic number in terms of L(G).

The remaining question asks whether a multiplicative bound might nevertheless hold: whether one universal constant can satisfy χ₌(G)≤C L(G) for every connected block graph G. The constructed graph G_{2,7} implies that any such constant would have to satisfy C≥9/7, while the family with k=4d−1 has ratio χ₌(G_{d,4d−1})/L(G_{d,4d−1}) tending to 5/4.